Waveform synthesis for imaging and ranging applications

ABSTRACT

Frequency dependent corrections are provided for quadrature imbalance. An operational procedure filters imbalance effects without prior calibration or equalization. Waveform generation can be adjusted/corrected in a synthetic aperture radar system (SAR), where a rolling phase shift is applied to the SAR&#39;s QDWS signal where it is demodulated in a receiver; unwanted energies, such as imbalance energy, are separated from a desired signal in Doppler; the separated energy is filtered from the receiver leaving the desired signal; and the separated energy in the receiver is measured to determine the degree of imbalance that is represented by it. Calibration methods can also be implemented into synthesis. The degree of quadrature imbalance can be used to determine calibration values that can then be provided as compensation for frequency dependent errors in components, such as the QDWS and SSB mixer, affecting quadrature signal quality.

STATEMENT OF GOVERNMENT INTEREST

The United States Government has rights in this invention pursuant toDepartment of Energy Contract No. DE-AC04-94AL85000 with SandiaCorporation.

FIELD OF THE INVENTION

The present invention is generally related to signal processing. Moreparticularly, the present invention is related to methods and systemsproviding waveform synthesis for imaging and ranging applications suchas RADAR, SONAR, LIDAR, medical imaging, tomography, and communicationsapplications utilizing spread spectrum modulation/demodulationtechniques.

BACKGROUND OF THE INVENTION

Although the present background describes the functionality andlimitations of synthetic aperture radar systems or a particular class ofcommunications, such description is merely provided to exemplify aproblem capable of resolution with the present invention. Any discussionherein directed to specific radars or communications protocols shouldnot be taken by those skilled in the art as a limitation on theapplicability of the invention described herein.

Modem high-performance radar systems often generate signals ofextraordinarily wide bandwidth. For example, the General Atomics LynxSynthetic Aperture Radar (SAR) employs a Linear-FM (LFM) chirp waveformand can operate over 3 GHz bandwidth at a 16.7 GHz center frequency.Furthermore, maximum exploitation of these radar signals requires thegenerated waveforms to be of very high quality, possessing exceptionalspectral purity.

To facilitate high-quality LFM chirp generation, a programmable DigitalWaveform Synthesizer (DWS) can often be employed. Use of a GaAs ASIC hasbeen shown to implement well known double accumulator architectures togenerate a phase that is quadratic with time, a phase that is generallyconverted by a memory look-up table to a digital sinusoidal signal andis ultimately converted to an analog signal by a Digital to AnalogConverter (DAC). Furthermore, an ability to predistort the phase of theoutput as a function of instantaneous frequency to correct forunspecified nonlinearities of subsequent components in the signal pathhas previously been explored. No calibration scheme, however, has beenpresented for determining correction factors.

Two principal architectures are presently employed for achieving LFMchirp generation. The first architecture 100 is referred to assingle-ended output operation, and is illustrated in FIG. 1 (labeled asprior art). With this architecture a single signal output is generatedby the DWS 110 and presented to subsequent components in the signalpath. After mixing 115 with a Local Oscillator (LO) 120 signal thenature of a single-ended DWS signal is to generate the desired signal aswell as an undesired mirror-image signal, which must be filtered byanalog components in the signal path. This filter is often called asideband filter 130. Consequently, desired and undesired signals areseparated by frequency; limiting the usable bandwidth for a generatedsignal to something less than half the DWS clock frequency.Equivalently, a clock frequency of more than twice the highest outputwaveform frequency is required. Proper final system bandwidth isachieved through a frequency multiplier 140.

Multiplexing of multiple parallel chirp generators can allow widebandwidth single-ended chirps to be generated with commercial siliconField Programmable Gate Array (FPGA) components. Frequencymultiplication can also be employed to widen the bandwidth of asingle-ended DWS output signal, but often to the detriment of spectralpurity. It is also well known that frequency multiplication raisesundesired frequency spurs by 6 dB per doubling with respect to thedesired signal level. Frequency spurs, however, are undesired signalperturbations caused by quantization effects and DAC residualnonlinearities. Consequently, minimizing the frequency multiplicationfactor that can be applied to a DWS output can enhance spectral purity.

A second architecture that has been used for achieving LFM chirpgeneration quality can be referred to as balanced or quadraturemodulator operation. Such architecture is generally illustrated in FIG.2 (also labeled as prior art). With this architecture 200, two outputsignals are generated by the Quadrature DWS (QDWS) 210 and presented toa Single Sideband (SSB) mixer 220 where they are combined 215 to form asingle signal to the subsequent signal path. In a perfect system, thetwo signals generated by the QDWS 210 will differ by a constant 90degrees of phase, and are termed the In-phase (I) and Quadrature-phase(Q) signals. The signal pair together can be generally referred to inthe art as Quadrature signals. In a perfect SSB mixer 220, nomirror-image signal will be generated, obviating the need for a sidebandfilter. Furthermore, no spectral separation between desired and anonexistent undesired signal would need to be maintained. Consequently,the QDWS 210 output bandwidth of the desired signal would be able toapproach the QDWS clock frequency itself, which is twice the bandwidthof the single-ended DWS system. This, in turn, would require half thefrequency multiplication 230 when compared to a single-ended DWS systemto achieve a final system bandwidth, and include attendant 6 dB lowerspur levels and better spectral purity.

SSB signal generation techniques, including the employment of quadraturesignals, are generally known in the art. Quadrature signals can begenerated by a variety of techniques, including Hilbert filters thatgenerate a 90-degree phase shift for all input waveform frequencies, anddirectly by separate memory look-up tables within the digital signalgeneration portion of the QDWS. The precision with which quadraturesignals can be generated and combined in a SSB mixer, however, isproblematic, particularly for high-dynamic-range applications such asimaging radar systems. Imperfections in quadrature signal generation ortheir combination within a SSB mixer results in the non-cancellation ofthe undesired mirror-image sideband signal. Such imperfections canresult in a relative phase error or an amplitude imbalance.Additionally, the LO 120 may undesirably leak through the mixer and bepresent in the mixer output in addition to the desired signal. Any ofthese errors reduce the spectral purity of the resulting SSB mixeroutput signal, and degrades a SAR image with ghosts and other artifacts.Consequently, quadrature modulators in high-performance radar systemsrequire some form of error cancellation or other mitigation scheme.

In the field of communications, a quadrature modulator for wireless CDMAsystems has been described wherein amplitude and phase of the quadraturecomponent signals are predistorted to provide perfect quadrature signalsto the SSB mixer. Furthermore, DC biases are added to the quadratureoutput signals to mitigate LO leakage. The corrections, however, arederived for a single QDWS output frequency and do not allow forfrequency dependent errors. While this may be reasonable forapplications such as wireless communications, it is inadequate forhigh-performance SAR systems. Furthermore, no attempt is made tocompensate imbalances in the SSB mixer itself, and the stated procedureprecludes this.

An iterative procedure to adjust for LO leakage, and phase and gainimbalance in a quadrature modulator based on an envelope detection of atransmitted signal has also been described by the prior art. Other priorart techniques include: use of a similar technique to compensate an SSBmixer which degrades a resultant output signal and adaptive techniquesfor achieving quadrature signal balance using a test tone. Theadaptation procedures currently described in the prior art is notadequate for wideband LFM chirps. Furthermore, verified widebandfrequency-dependent errors are not addressed by the prior art.

A quadrature modulator that allows frequency-dependent phase andamplitude corrections to be made to the output of the QDWS has beenproposed. All corrections are made to analog signals after the DACs.These corrections, however, neglect problematic frequency dependenterrors in the SSB mixer. Furthermore, the nature of the errors to becorrected is presumed to be predetermined—that is, no calibrationprocedure is discussed.

A quadrature modulator has been described that can be constructed thatimplements frequency dependent corrections before and at the DACs. TheSSB mixer, however, is not addressed, nor is correction due to imbalancefrom any components or sources addressed.

Thus far, quadrature modulator prior art generally is inadequate forhigh-performance radar systems for one or more of the following reasons:

1) The balance corrections are relatively narrowband, and do notfacilitate frequency dependent phase and amplitude corrections.

2) Adaptive schemes are not fast enough for LFM wideband chirpwaveforms.

3) Imbalances within the SSB mixer itself are not addressed.

4) LO leakage (e.g., especially any frequency dependence) is notadequately addressed.

5) Where calibration schemes should be required, no calibration schemesare proposed.

The prior art leaves the skilled in the art with limited choices betweennarrow-band quadrature modulators that can compensate for imbalances andLO leakage using iterative techniques and wideband quadrature modulatorsthat do not compensate for frequency dependent errors in the SSB mixer.Accordingly, there is a need for an improved wideband quadraturemodulator and SSB mixer combination, which produces suitable signals foruse in high fidelity radar waveform generation. Additionally, there isalso a need for an operational methodology that is insensitive to theimbalances and leakage in a quadrature modulator and associated SSBmixer.

SUMMARY OF THE INVENTION

In the field of quadrature demodulators, which are used in radarreceivers, U.S. Pat. No. 6,469,661, issued Oct. 22, 2002 to two of thepresent inventors teaches how errors due to I and Q channel imbalancesin a receiver's demodulator can be mitigated by processing radar signalswith pulse-to-pulse, rolling phase shifts. With this technique nocalibration is required to achieve a balanced demodulator output signal.The use of quadrature modulators in radar transmitters, however, has notbeen described. The present inventors have discovered how use of aQuadrature DWS is viable for high-performance radar waveform synthesis.A principal advantage of the present invention is that fewer frequencymultipliers are needed to generate radar waveforms of sufficientbandwidth; thereby offering the potential of cleaner waveforms (e.g.,signals processing lower frequency-spurs).

Problems that generally need to be resolved with QDWS signals are thoserelated to the generation of unwanted signal components due to energiessuch as LO feed-through found in the non-ideal SSB mixer and quadraturesignal imbalance, as well as problem typically caused within the SSBmixer. For example, the inclusion of unwanted energy into a radar'soutput (e.g. SAR images) creates problems associated with the QDWS andSSB components that result in signal waveform degradation, and henceradar performance. Similar problems are encountered with other systemsengaged in or requiring signal/waveform synthesis, such as: SONAR,LIDAR, medical imaging, tomography, and communications.

It is a feature of the present invention to provide an improvedQuadrature Digital Waveform Synthesizer that can further provide systemswith frequency dependent corrections for quadrature imbalance and LocalOscillator (LO) feed-through.

It is another feature of the present invention to provide operationalprocedures to system that can enable filtration of quadrature imbalanceeffects from energies, such as LO feed-through energy and imbalanceenergy, and their effects without prior calibration or equalization.

It is yet another feature of the present invention to providecalibration procedures that can also be implemented into systems forsignal synthesis.

Techniques that are herein disclosed by the present inventors tomitigate the negative effects of signal components brought on byunwanted energies such as LO feed-through and quadrature signalimbalance include the benefits of:

1) Compensating or equalizing energy imbalance by adjusting inputs (DCoffsets, phase, and amplitude) to the SSB mixer so that problematicsignal components aren't generated, and

2) Operating in a manner (e.g. by employing rolling phase shifts) thatseparates problem signals (i.e., energy) from a desired signal inDoppler to allow filtering to suppress problem energy.

In accordance with methods of the present invention, waveform generationin a system can be adjusted/corrected, for example, in a syntheticaperture radar system (SAR), where a rolling phase shift is applied tothe SAR's QDWS signal where it is demodulated in a receiver; the LOfeed-through energy and/or imbalance energy are then separated from adesired signal in Doppler; the separated energy is filtered from thereceiver leaving the desired signal; and the separated energy in thereceiver is measured to determine the degree of imbalance that isrepresented by it. The degree of imbalance in the system can be used todetermine calibration values that can then be provided as compensationfor frequency dependent errors in components, such as a SAR's QDWS andSSB mixer, affecting quadrature signal quality.

Error correction circuits can be built into the QDWS that compensatewideband frequency dependent phase and amplitude imbalances originatingin the QDWS, SSB mixer, and other non-ideal components in the I and Qchannels of the quadrature modulator. This can be done by predistortingthe relative phase and amplitudes of the QDWS quadrature signalcomponents by programmable frequency dependent digital perturbationsinto the QDWS prior to analog signal generation. LO leakage in the SSBmixer is thereafter compensated by applying DC offsets to the SSBinputs.

Frequency dependent phase and amplitude corrections and LO leakagecompensation can be determined by a calibration process that involvesapplying pulse-to-pulse rolling phase shifts to the LFM signal withinthe QDWS. These phase shifts can be removed in the receiver by conjugatephase shifts. The application of these phase shifts can separate desiredsignals from differential-mode imbalance energy in a Doppler spectrum(the spectrum taken across the pulses). Separation allows for the uniqueidentification of the nature of the quadrature imbalance and LO leakageas a function of frequency for calibration purposes.

Alternatively, the present inventors teach that the calibration processcan be dispensed with, where Doppler filtering can be applied to thereceived data to simply remove the imbalance, such as LO leakage energy,from the received signal. Calibration can thereby be renderedunnecessary, rendering any quadrature signal imbalances, or LO leakage,impotent.

BRIEF DESCRIPTION OF THE DRAWINGS

The accompanying drawings, which are incorporated in and form part ofthe specification, illustrate embodiments of the present invention and,together with the description, serve to explain relevant principles ofthe invention.

FIG. 1 provides a prior art illustration of a single-ended DWS withfrequency conversion, sideband filter, and frequency multiplicationcomponents.

FIG. 2 provides a prior art illustration of a quadrature modulatoremploying a QDWS, SSB mixer, and frequency multiplier.

FIG. 3 illustrates a non-ideal SSB Mixer model.

FIG. 4 illustrates a simplified radar-processing model.

FIG. 5 illustrates a simplified non-ideal SSB Mixer model.

FIG. 6 illustrates a simplified non-ideal SSB Mixer model with LOfeed-through compensation.

FIG. 7 illustrates a Simplified non-ideal SSB Mixer model with LOfeed-through compensation and quadrature signal equalization.

FIG. 8 illustrates a LFM chirp generation using QECDWS. The PEC RAMcontains phase corrections common to both I and Q channels. The QECphase RAM corrects phase differences, and the QEC magnitude RAM correctsamplitude differences.

FIG. 9 illustrates a Frequency representation of a conventionaltechnique for generating QDWS output.

FIG. 10 illustrates the generation of QDWS output with frequency bias.

FIG. 11 illustrates the generation of QDWS output with frequency bias.

FIG. 12 illustrates filtering effects of non-ideal DAC.

FIG. 13 illustrates Doppler separation of LO feed-through signal fromdesired signal using π rad./pulse modulation of the QDWS phase.

FIG. 14 illustrates a method for generating π rad./pulse modulation ofthe QDWS phase.

FIG. 15 illustrates Doppler separation of LO feed-through signal and I/Qimbalance signal from desired signal using rolling phase shift of theQDWS phase.

FIG. 16 illustrates an example energy map (image) of point scatterer inradar with LO feed-through and I/Q imbalance.

FIG. 17 illustrates insertion of a rolling phase shift into the QDWSoutput.

FIG. 18 illustrates a method for generating π/2 rad./pulse modulation ofthe QDWS phase.

FIG. 19 illustrates a simulation of gradient search method for nullingLO feed-through signal, SNR is for desired signal after compression.

FIG. 20 illustrates the relationship of residual video LFM chirp due toLO feed-through to video bandwidth and relative echo delay time.

FIG. 21 illustrates the relationship of residual video LFM chirp due toquadrature channel imbalance to video bandwidth and relative echo delaytime.

FIGS. 22A-22D illustrate simulations of LO feed-through migration andI/Q imbalance mitigation using gradient search parameter estimation.Images displayed are periodic snapshots of iterative cycles.

DETAILED DESCRIPTION OF THE INVENTION

Although the following provides a detailed description of thefunctionality and limitations that can be found in synthetic apertureradar (SAR) systems or may refer to a particular class ofcommunications, such description is merely provided to exemplify asignaling problem commonly found in imaging, ranging and communicationequipment that is capable of resolution with the present invention. Anydiscussion herein directed to specific radars or communications andtheir associated protocols should not be taken by those skilled in theart as a limitation on the applicability of the invention describedherein.

The following detailed description is comprised of three main sections:

A. A model for errors and their impact on radar data, including errorsdue to the SSB mixer, effects of multipliers, and radar processing.

B. Mitigation techniques for non-ideal performance, includingequalization techniques and operational techniques that do not requirecalibration.

C. Parameter estimation for equalization, including procedures forcalibrating the QDWS.

A. A Model for Errors and their Impact on Radar Data

FIG. 3 illustrates an architecture 300 for the non-ideal mixer. Theeffects of non-ideal QDWS, SSB mixer, and related components can bemodeled in the following development:

Module H_(I)(f)=the application of the perturbations offered to the Ichannel 310,

Module H_(Q)(f)=the application of the perturbations offered to the Qchannel 320,

Module H_(O)(f)=the application of the perturbations offered to the LOsignal 330, and

Module H_(L)(f)=the transfer function of the LO leakage signal to theoutput 340.

When the Local Oscillator is a single-frequency sinusoid with frequencyω₀=2πf₀, the transfer functions H_(O)(f) and H_(L)(f) represent constantperturbations to amplitude and phase. Consequently, for thesingle-frequency LO, intermediate signals can be identified as:

i ₀=cos(ω₀ t),

q ₀=(1+k _(q0))sin(ω₀ t+θ _(q0)) and

g ₀ =k _(g0) cos(ω₀ t+θ _(g0)).

For ideal quadrature input signals, k_(q0)=k_(g0)=0, and θ_(q0)=θ_(g0)=0are desired.

Correspondingly, the transfer functions can be identified by H_(I)(f)and H_(Q)(f) as descriptions of frequency-dependent amplitude and phaseperturbations to their respective quadrature signal inputs. Thesequadrature signals would have been originated by the QDWS.

Consider now the inputs to be defined as single frequency sinusoids:

x _(QDWS,I)(t)=cos(ω_(T) t), and

x _(QDWS,Q)(t)=sin(ω_(T) t).

In a radar waveform, these signals are normally finite in duration. Theenvelope in the following development, however, can be ignored tosimplify the discussion. Nevertheless, an underlying presumption isgenerally that the envelope does in fact exist, and will need to bereinserted at some point.

The corresponding intermediate signals would then be identified as:

i ₁=(1+k _(i1))cos(ω_(T) t+θ _(i1)), and

q ₁=(1+k _(q1))sin(ω_(T) t+θ _(q1)).

Again, if the applied quadrature signals were ideal, k_(i1)=k_(q1)=0 andθ_(i1)=θ_(q1)=0 would be desirable. If not equal to zero, the next bestcondition can be that the respective parameters are equal to each other,to at least maintain balance. In any case, in a practical system weexpect these perturbations to be frequency dependent.

Multiplying out the intermediate signals yields: $\begin{matrix}{{x_{Mixer}(t)} = {{i_{0}i_{1}} - {q_{0}q_{1}} + g_{0}}} \\{= {\begin{pmatrix}{{\left( {1 + k_{i1}} \right){\cos \left( {\omega_{0}t} \right)}{\cos \left( {{\omega_{T}t} + \theta_{i1}} \right)}} -} \\{{\left( {1 + k_{q0}} \right)\left( {1 + k_{q1}} \right){\sin \left( {{\omega_{0}t} + \theta_{q0}} \right)}{\sin \left( {{\omega_{T}t} + \theta_{q1}} \right)}} +} \\{k_{g0}{\cos \left( {{\omega_{0}t} + \theta_{g0}} \right)}}\end{pmatrix}.}}\end{matrix}$

It can be noted that as far as the output is concerned, a scaling errork_(q0) is indistinguishable from a scaling error k_(q1). That is, forevery perturbation in one, a corresponding perturbation exists in theother. Consequently, no generality can be lost by assuming k_(q0)=0.This yields: ${x_{Mixer}(t)} = {\begin{pmatrix}{{\left( {1 + k_{i1}} \right){\cos \left( {\omega_{0}t} \right)}{\cos \left( {{\omega_{T}t} + \theta_{i1}} \right)}} -} \\{{\left( {1 + k_{q1}} \right){\sin \left( {{\omega_{0}t} + \theta_{q0}} \right)}{\sin \left( {{\omega_{T}t} + \theta_{q1}} \right)}} +} \\{k_{g0}{\cos \left( {{\omega_{0}t} + \theta_{g0}} \right)}}\end{pmatrix}.}$

The phase perturbations θ_(q0), however, is not indistinguishable fromany other term. Consequently, θ_(q0)=0 cannot be assumed and must becarried along.

For small imbalances and small perturbations, the mixer output can beexpanded and approximated as: ${x_{Mixer}(t)} = {\begin{pmatrix}{{\left\lbrack {\left( {1 + k_{i1}} \right) + \frac{\left( {K_{q1} - k_{i1}} \right)}{2}} \right\rbrack {\cos \left( {{\omega_{0}t} + {\omega_{T}t} + \theta_{i1}} \right)}} -} \\{{\left\lbrack \frac{\theta_{q1} + \theta_{q0} - \theta_{i1}}{2} \right\rbrack {\sin \left( {{\omega_{0}t} + {\omega_{T}t} + \theta_{i1}} \right)}} -} \\{{\left\lbrack \frac{k_{q1} - k_{i1}}{2} \right\rbrack {\cos \left( {{\omega_{0}t} - {\omega_{T}t} - \theta_{i1}} \right)}} -} \\{{\left\lbrack \frac{\theta_{q1} - \theta_{q0} - \theta_{i1}}{2} \right\rbrack {\sin \left( {{\omega_{0}t} - {\omega_{T}t} - \theta_{i1}} \right)}} +} \\{k_{g0}{\cos \left( {{\omega_{0}t} + \theta_{g0}} \right)}}\end{pmatrix}.}$

This signal is considered real valued. Consequently, spectral symmetryallows us to work with just the positive frequencies. Positivefrequencies can be extracted and manipulated to the form:${{x_{Mixer}(t)} = {\frac{1}{2}\exp \quad {j\left( {\omega_{0}t} \right)}\begin{pmatrix}{{\left\lbrack {1 + k_{i1} + \frac{\Delta \quad k}{2} + {j\left( {\frac{\Delta\theta}{2} + \theta_{q0}} \right)}} \right\rbrack \exp \quad {j\left( {{\omega_{T}t} + \theta_{i1}} \right)}} -} \\{{\left\lbrack {\frac{\Delta \quad k}{2} - {j\frac{\Delta\theta}{2}}} \right\rbrack \exp} - {j\left( {{\omega_{T}t} + \theta_{i1}} \right)} +} \\{k_{g0}\exp \quad j\quad \theta_{g0}}\end{pmatrix}}},$

where the imbalances have been defined as (k_(q1)−k_(i1))=Δk, and(θ_(q1)−θ_(q0)−θ_(i1))=Δθ. In this form, it should be noted that thecarrier is modulated by three terms, each with distinctly differentcharacteristics.

If, ideally, no imbalances exist, this would reduce to elements of thefirst modulation term, that is:${x_{{Mixer},\quad {{No}\quad {Imbalances}}}(t)} = {{\frac{1}{2}\left\lbrack {1 + k_{i1} + {j\quad \theta_{q0}}} \right\rbrack}\exp \quad j\quad {\left( {{\omega_{0}t} + {\omega_{T}t} + \theta_{i1}} \right).}}$

No perturbations of any kind would further reduce this to:${x_{{Mixer},\quad {Ideal}}(t)} = {\frac{1}{2}\exp \quad {{j\left( {{\omega_{0}t} + {\omega_{T}t}} \right)}.}}$

The first modulation term in x_(Mixer)(t) is desired, and for smallimbalances is very nearly ideal, in the relative sense.

The second modulation term is due to imbalances, and contains theconjugate, or “image” of the desired modulation.

The third term is due to local oscillator feed-through, which manifestsitself as a DC modulation of the carrier, albeit with some phase shift.

The effects of imbalance to the magnitude and phase of the desired termare typically inconsequential. Consequently, the mixer output can befurther simplified to:${x_{Mixer}(t)} = {\frac{1}{2}\exp \quad {j\left( {\omega_{0}t} \right)}{\begin{pmatrix}{{\left\lbrack {1 + k_{i1} + {j\quad \theta_{q0}}} \right\rbrack \exp \quad {j\left( {{\omega_{T}t} + \theta_{i1}} \right)}} -} \\{{\left\lbrack {\frac{\Delta \quad k}{2} - {j\frac{\Delta\theta}{2}}} \right\rbrack \exp} - {j\left( {{\omega_{T}t} + \theta_{i1}} \right)} +} \\{k_{g0}\exp \quad j\quad \theta_{g0}}\end{pmatrix}.}}$

Furthermore, if ω_(T) is allowed to vary with time itself, as with a LFMchirp, ω_(T)t can be replaced with the more general phase function Φ(t),but the fact that Δk and Δθ are frequency dependent should be kept inmind. The mixer output can then be described as:${x_{Mixer}(t)} = \begin{pmatrix}{A_{0}\exp \quad {j\left( {{\omega_{0}t} + {\Phi (t)} + \theta_{i1} + \phi_{0}} \right)} \times} \\\begin{pmatrix}{1 +} \\{{A_{I}\exp} - {j\left( {{2{\Phi (t)}} + {2\theta_{i1}} + \phi_{I}} \right)} +} \\{{A_{L}\exp} - {j\left( {{\Phi (t)} + \theta_{i1} - \theta_{g0} - \phi_{L}} \right)}}\end{pmatrix}\end{pmatrix}$

where frequency dependent terms can be defined as:${A_{0}\exp \quad j\quad \phi_{0}} = \left( \frac{1 + k_{i1} + {j\quad \theta_{q0}}}{2} \right)$${{{A_{1}\exp} - {j\quad \phi_{1}}} = {{- \frac{1}{2}}\left( \frac{{\Delta \quad k} - {j\quad {\Delta\theta}}}{1 + k_{i1} + {j\quad \theta_{q0}}} \right)}},{and}$${A_{L}\exp \quad j\quad \phi_{L}} = {\left( \frac{k_{g0}}{1 + k_{i1} + {j\quad \theta_{q0}}} \right).}$

The frequency multiplier output can initially take this expression tothe power M, and then applies appropriate filtering. Furthermore, thenon-linear compression nature of the multiplication stages typicallymitigates to some extent any absolute amplitude fluctuations. For smallimbalances, the net result is then reasonably modeled as:${x_{M}(t)} \approx {\begin{pmatrix}{{\exp \quad {j\left( {{M\quad \omega_{0}t} + {M\quad {\Phi (t)}} + {M\quad \theta_{i1}} + {M\quad \phi_{0}}} \right)} \times}\quad} \\\begin{pmatrix}{1 +} \\{{{MA}_{I}\exp} - {j\left( {{2{\Phi (t)}} + {2\theta_{i1}} + \phi_{1}} \right)} +} \\{{{MA}_{L}\exp} - {j\left( {{\Phi (t)} + \theta_{i1} - \theta_{g0} - \phi_{L}} \right)}}\end{pmatrix}\end{pmatrix}.}$

The increase in the level of the undesired terms by a factor M is nowevident. As this happens to all pre-multiplier spurs, it rationalizes adesire to minimize the multiplication factor in an overall systemdesign.

The Radar Receiver Video Signal and Correlator Output

For purposes of discussion, a simplified radar system 400 is illustratedin FIG. 4. Output from the multiplier 401 can be presumed to be a radartransmitted (TX) signal. The received (RX) echo signal is delayable 410by an amount t_(s). The receiver can mix 415 echo with a receiver localoscillator signal 420 in the form:

x _(L)(t)=x _(M)*(t−t _(m))

to form a video signal 421. The superscript * denotes complex conjugate.

The video signal can be written in a reduced form (ignoring somenegligible terms) as: ${x_{V}(t)} = {\begin{bmatrix}{\exp \quad {j\left( {{M\quad {\omega_{0}\left( {t_{m} - t_{s}} \right)}} + {M\left( {{\Phi \left( {t - t_{s}} \right)} -} \right.}} \right.}} \\{\left. {\left. {\Phi \left( {t - t_{m}} \right)} \right) + {M\left( {\phi_{0,s} - \phi_{0,m}} \right)}} \right) \times} \\\begin{pmatrix}{1 +} \\{{MA}_{I,m}\exp \quad {j\left( {{2{\Phi \left( {t - t_{m}} \right)}} + {2\theta_{{i1},m}} + \phi_{I,m} +} \right.}} \\{{{MA}_{I,s}\exp} - \quad {j\left( {{2{\Phi \left( {t - t_{s}} \right)}} + {2\theta_{{i1},s}} + \phi_{I,s} +} \right.}} \\{{{MA}_{L,m}\exp \quad {j\left( {{\Phi \left( {t - t_{m}} \right)} + \theta_{{i1},m} - \theta_{{g0},m} - \phi_{L,m}} \right)}} +} \\{{{MA}_{L,s}\exp} - \quad {j\left( {{\Phi \left( {t - t_{s}} \right)} + \theta_{{i1},s} - \theta_{{g0},s} - \phi_{L,s}} \right)}}\end{pmatrix}\end{bmatrix}.}$

It should be noted that a number of terms are in fact frequencydependent—that is, dependent on the instantaneous frequency of eitherΦ(t−t_(s)) or Φ(t−t_(m)) which may in fact be different from each otherat any instant in time. Consequently additional subscripts have beenadded herein to distinguish these terms as being influenced by whichinstantaneous frequency.

Strictly speaking, because pulses are being dealt with, unlesst_(m)=t_(s), there will typically be some non-overlap in the envelopesof the two signals. This is generally deemed normal for “stretchprocessing” and can be compensated by lengthening the envelopes (pulsewidths) slightly beyond the interval over which data is typicallycollected from the video signal. Nevertheless, development can proceedas if the envelopes do in fact overlap sufficiently. The envelope of avideo signal itself is adequately represented by:${{{rect}\left( \frac{t - t_{m}}{T} \right)}{{rect}\left( \frac{t - t_{s}}{T} \right)}} \approx {{rect}\left( \frac{t - t_{m}}{T} \right)}$

where T is the pulse width.

The phase term M(φ_(0,s)−φ_(0.m)) represents undesired phase variationsin the desired video component, due to non-ideal signal paths. Theresult in SAR images, for example, is range-sidelobes, even where I andQ channels are completely balanced and LO feed-through entirelysuppressed. This is precisely the term that more traditional DWSPhase-Error Correction is targeted to fix.

The integral of the video signal over time is the correlator output forthe radar and describes the Impulse Response (IPR) of the radar. Acorrelation peak occurs when t_(m)=t_(s), for which the video signal is:${{x_{V}(t)}}_{t_{m} = t_{s}}{\begin{pmatrix}{1 +} \\{{2{MA}_{I,s}\cos \quad \left( {{2{\Phi \left( {t - t_{s}} \right)}} + {2\theta_{{i1},s}} + \phi_{I,s}} \right)} +} \\{2{MA}_{L,s}\cos \quad \left( {{\Phi \left( {t - t_{s}} \right)} + \theta_{{i1},s} - \theta_{{g0},s} - \phi_{L,s}} \right)}\end{pmatrix}.}$

Several observations can be noted:

The “1” describes the properly focused signal that would be sought.

The second term is the residual due to imbalance in the I and Q channelsof the QDWS and SSB mixer. This signal retains time dependence, withfrequency components of 2Φ(t)—that is, its energy is spread across twicethe bandwidth of Φ(t).

The third term is the residual due to LO feed-through in the SSB mixer.This signal also retains time dependence, with frequency components ofΦ(t). That is, its energy is spread across just the bandwidth of Φ(t).

The second and third terms are undesired and constitute ‘multiplicativenoise’ in that their energy level is proportional to the desiredcomponent's response energy level. The total multiplicative noisegenerated relative to the signal is then:

MNR _(DWS)=2M ²(A _(I) ² +A _(L) ²)

where some sort of average value of A_(I) and A_(L) are used.

While this expression indicates the total relative undesired energyavailable, it fails to account for any diminishment due to videopass-band filtering, and may in some circumstances be too pessimistic.The video signal will normally have a bandwidth much reduced from thatof x_(M)(t). It can be noted, however, that undesired terms arefunctions whose phase are themselves time-dependent—that is, whose phasedepends on delayed versions of Φ(t). Typical phase functions Φ(t) forradar are such that exp jΦ(t) has a spectral bandwidth (as with a LFMchirp), that when appropriately multiplied, allows a desired rangeresolution. Consequently, if we denote B_(Φ) as the bandwidth of expjΦ(t), then energy due to I and Q channel imbalance is spread over abandwidth 2B_(Φ), and energy due to LO feed-through is spread over abandwidth B_(Φ).

If the video signal is filtered (as it typically is) to some bandwidthB_(V), then some of the offending undesired energy may be filtered, andrendered unavailable to the correlator output. Consequently, fractionalbandwidths can be defined as:${\beta_{I} = {\frac{2B_{V}}{2B_{\Phi}} = \frac{{MB}_{V}}{B_{M}}}},{and}$$\beta_{L} = {\frac{2B_{V}}{B_{\Phi}} = {\frac{2{MB}_{V}}{B_{M}}.}}$

but limited to β_(I)≦1 and β_(L)≦1, and with B_(M) the bandwidth of theTX signal x_(M)(t) as previously defined. With these definitions, theMultiplicative Noise Ratio (MNR) from a single return echo is limited to

MMR _(DWS)=2M ²(β_(I) A _(I) ²+β_(L) A _(L) ²).

As an example, given M=8, B_(V)=30 MHz, and B_(M)=1800 MHz for 0.1-mresolution, would result in β_(I)=0.133, and β_(L)=0.267, offering 8.75dB and 5.74 dB reductions in the MNR due to their respective sources.The reduction in MN, however, may not always be as much as one mightthink. For example, given M=6, B_(V)=30 MHz, and B_(M)=180 MHz for 1-mresolution, would result in β_(I)=β_(L)=1 with no MNR reduction due tothe reduced video filter. Note that higher multiplication factors M, andcoarser resolutions render the video filter less effective in reducingMNR due to imbalance and LO feed-through.

An additional problem can be found when the bandwidth of the undesiredsignals extends beyond the video filter bandwidth, which also impliesthat radar echoes that would ordinarily be filtered by the video filtermight still throw energy into the video pass band due to LO feed-throughand I/Q imbalance.

In any case, for a versatile high-performance radar system, the videofilter cannot be relied upon to adequately reduce the effects ofimbalance and LO feed-through. Consequently, the imbalance and LOfeed-through (or at least the effects thereof) generally need to bemitigated by other techniques.

B. Mitigation Techniques for Non-ideal Performance

Techniques for dealing with problematic QDWS and SSB mixer associatedI/Q channel imbalance and LO feed-through can generally fall into threebroad classes:

1) Compensating or equalizing the imbalance and LO feed-through so thatproblematic signal components aren't generated,

2) Separating the problematic signal components from the desired signalin a manner that allows filtering to work, and

3) Dealing with the effects of the problematic signal components in thereceiver.

Of course, in light of the present description, one should appreciatethat various ideas can be mixed and matched from all three classes intoan overall strategy to maximize system performance.

Equalization Techniques

The first mitigation technique that will now be described involvesequalizing the quadrature channels. One goal of this technique is tocause the SSB mixer output to be as nearly ideal as possible. This canbe achieved by pre-distorting the SSB input signal in a complementarymanner from its own distortion, such that the net result is an idealsignal at the mixer output.

There are generally two distinct problem signals. The first is due to LOfeed-through, and the second is due to I/Q channel imbalance. Each canbe treated in order. To facilitate this development the architecture ofan SSB mixer 520 itself can be desired to, as is illustrated in thesystem 500 of FIG. 5, where one can appreciate the mathematical model:${x_{Mixer}(t)} = {\begin{pmatrix}{A_{0}\exp \quad {j\left( {{\omega_{0}t} + {\Phi (t)} + \theta_{i1} + \phi_{0}} \right)} \times} \\\begin{pmatrix}{1 +} \\{{A_{1}\exp} - {j\left( {{2{\Phi (t)}} + {2\theta_{i1}} + \phi_{1}} \right)} +} \\{{A_{L}\exp} - {j\left( {{\Phi (t)} + \theta_{i1} - \theta_{g0} - \phi_{L}} \right)}}\end{pmatrix}\end{pmatrix}.}$

Also, it should be noted that the effects of k_(q0) have been absorbedinto k_(q1), and hence A_(I) and Φ_(I). The goal here is to make thequantities A_(I) and A_(L) as small as possible, ideally zero.

Compensating LO Feed-through

The LO feed-through energy may be mitigated by forcing A_(L)=0 byappropriately modifying or selecting properties of x_(QDWS,I)(t) andx_(QDWS,Q)(t). Note that the LO feed-through signal can manifest itselfas an equivalent DC modulation of the LO signal. By forcing inputsignals x_(QDWS,I)(t) and x_(QDWS,Q)(t) to be DC signals, withrespective values:

 x _(QDWS,I)(t)=V _(L,I), and

x _(QDWS,Q)(t)=V _(L,Q),

intermediate signals can be identified as:

i ₁ =V _(L,I)H_(I)(0), and

q ₁ =V _(L,Q) H _(Q)(0).

For the DC signal input, the SSB mixer 520 output can then be describedby: ${x_{Mixer}(t)} = {\begin{pmatrix}{{\left\lfloor {{V_{L,I}{H_{I}(0)}} + {k_{g0}\cos \quad \left( \theta_{g0} \right)} - {V_{L,Q}{H_{Q}(0)}{\sin \left( \theta_{q0} \right)}}} \right\rfloor \cos \quad \left( {\omega_{0}t} \right)} -} \\{\left\lbrack {{V_{L,Q}{H_{Q}(0)}\cos \quad \left( \theta_{q0} \right)} + {k_{g0}\sin \quad \left( \theta_{g0} \right)}} \right\rbrack \sin \quad \left( {\omega_{0}t} \right)}\end{pmatrix}.}$

Positive frequencies can be described by:${x_{Mixer}(t)} = {\frac{1}{2}\begin{pmatrix}{\left\lfloor {{V_{L,I}{H_{I}(0)}} + {k_{g0}\cos \quad \left( \theta_{g0} \right)} - {V_{L,Q}{H_{Q}(0)}{\sin \left( \theta_{q0} \right)}}} \right\rfloor +} \\{j\left\lbrack {{V_{L,Q}{H_{Q}(0)}\cos \quad \left( \theta_{q0} \right)} + {k_{g0}{\sin \left( \theta_{g0} \right)}}} \right\rbrack}\end{pmatrix}\exp \quad {{j\left( {\omega_{0}t} \right)}.}}$

The mixer output could be zeroed, thereby mitigating the effects of LOfeed-through, by setting:${V_{L,Q} = {- \left( \frac{k_{g0}\sin \quad \left( \theta_{g0} \right)}{{H_{Q}(0)}\cos \quad \left( \theta_{q0} \right)} \right)}},{and}$$V_{L,I} = {- {\left( \frac{{k_{g0}{\cos \left( \theta_{g0} \right)}} - {V_{L,Q}{H_{Q}(0)}\sin \quad \left( \theta_{q0} \right)}}{H_{I}(0)} \right).}}$

Clearly, as long as these DC values are properly input to the SSB mixer520, the LO feed-through can be nullified, that is A_(L)=0, even in thepresence of other spectral components. FIG. 6 illustrates a system 600wherein DC biases 610 can be added to the SSB mixer 620 input toaccomplish compensation. It can be noted also that θ_(q0)≠0 can causethe appropriate value for V_(L,I) to depend on the proper value forV_(L,Q). If the LO feed-through amplitude and phase are dependent on thefrequency of the input signal x_(QDWS)(t), then the DC compensationvalues V_(L,I) and V_(L,Q) need to be dependent on the instantaneousfrequency generated by a QDWS.

Balancing I/Q Channels with Error Correction

The quadrature channels can be balanced by forcing A_(I)=0 byappropriately modifying or selecting x_(QDWS,I)(t) and x_(QDWS,Q)(t). Inthis aspect of the invention the imbalance can be found in the transferfunctions H_(I)(f) and H_(Q)(f), and LO phase imbalance term θ_(q0).

To compensate for the undesired transfer functions H_(I)(f) andH_(Q)(f), and phase imbalance θ_(q0), equalization filters 710 can beinserted between the DWS output and the SSB mixer 720, as illustratedfor the system 700 shown in FIG. 7. Alternately the behavior of theequalization filters 710 can be incorporated into the QDWS itself bysuitably perturbing the quadrature signals.

A model for the input signals can be returned to, namely:

x _(QDWS,I)(t)=cos(Φ(t)), and

x _(QDWS,Q)(t)=sin(Φ(t))

and the limiting envelope rect(t/T) can continue to be ignored. In theabsence of the equalization filters 720 (and ignoring the DC offsetinputs), we recall the intermediate signals to be:

 i ₁=(1+k _(i1))cos(Φ(t)+θ_(i1)), and

q ₁=(1+k _(q1))sin(Φ(t)+θ_(q1))

where perturbation parameters k_(i1), θ_(i1), k_(q1), and θ_(q1) embodythe effects of H_(I)(f) and H_(Q)(f), and depend on (may vary with) theinstantaneous frequency ω_(T)=dΦ(t)/dt.

The effects of the equalization filters 710 can now be added, and itshould be noted that these transfer functions are also just descriptionsof amplitude and phase variations as a function of frequency.Consequently the effects of the equalization filters 710 can bemathematically modeled as:

i ₁=(1+k _(i1,Heq))(1+k _(i1))cos(Φ(t)+θ_(i1)+θ_(i1,Heq)), and

q ₁=(1+k _(q1,Heq))(1+k _(q1))sin(Φ(t)+θ_(q1)+θ_(q1,Heq))

where k_(i1,Heq), θ_(i1,Heq), k_(q1,Heq), and θ_(q1,Heq) describe theeffects of the equalization filters 710 at frequency ω_(T).

For the subsequent development equivalent parameters can now simply bedefined as:

(1+k _(i1,eq))=(1+k _(i1,Heq))(1+k _(i1)),

θ_(i1,eq)=θ_(i1)+θ_(i1,Heq),

(1+k _(q1,eq))=(1+k _(q1,Heq))(1+k _(q1)), and

The mixer 720 output without any LO feed-through, can also be describedby: ${x_{Mixer}(t)} = {\begin{pmatrix}{A_{0}\exp \quad {j\left( {{\omega_{0}t} + {\Phi (t)} + \theta_{{i1},{eq}} + \phi_{0}} \right)} \times} \\\left( {1 + {A_{1}\exp} - {j\left( {{2\quad {\Phi (t)}} + {2\quad \theta_{{i1},{eq}}} + \phi_{1}} \right)}} \right)\end{pmatrix}.}$

with the grouped parameters now defined by:${A_{0}\exp \quad j\quad {\phi_{0}\left( \frac{1 + k_{{i1},{eq}} + {j\quad \theta_{q0}}}{2} \right)}},{{{A_{1}\exp} - {j\quad \phi_{1}}} = {{- \frac{1}{2}}\left( \frac{{\Delta \quad k} - {j\quad \Delta \quad \theta}}{1 + k_{{i1},{e1}} + {j\quad \theta_{q0}}} \right)}},$

where now Δk=(k_(q1,eq)−k_(i1,eq)), and Δθ=(θ_(q1,eq)−θ_(i1,eq)−θ_(q0)).

To force A_(I)=0, Δk=Δθ=0 is required and further requires:${k_{{q1},{Heq}} = {{\left( {1 + k_{{i1},{Heq}}} \right)\frac{\left( {1 + k_{i1}} \right)}{\left( {1 + k_{q1}} \right)}} - 1}},{and}$

 θ_(q1,Heq)=θ_(i1)−θ_(q1)+θ_(i1,Heq)+θ_(q0).

k_(i1,Heq)=θ_(i1,Heq)=0 can be arbitrarily set and still maintaincomplete balance, or complete equalization can be forced by setting:${k_{{i1},{Heq}} = {\frac{1}{\left( {1 + k_{i1}} \right)} - 1}},{and}$

 θ_(i1,Heq)=−θ_(i1).

This would force k_(q1,eq)=k_(i1,eq)=0, andθ_(i1,eq)=(θ_(q1,eq)−θ_(q0))=0.

Equalization can therefore be accomplished for LFM chirp generation byimplementing the required amplitude and phase corrections of theequalization filters 710 directly into the QDWS itself. This schemewould produce x_(QDWS,I)(t) and x_(QDWS,Q)(t) with the error correctionsalready “built in”, and can be referred to as Quadrature Error CorrectedDWS (QECDWS). A block diagram detailing how this might be incorporatedis illustrated for the system architecture 800 shown in FIG. 8.

In the system architecture 800, the frequency accumulator 810 and phaseaccumulator 820 constitute the double accumulator architecture for LFMchirp generation. The PEC RAM 830 applies frequency dependentcommon-mode phase error corrections of the type previously discussed.The QEC Phase RAM 840 applies frequency dependent correctionsθ_(q1,Heq)=θ_(i1)−θ_(q1)+θ_(i1,Heq)+θ_(q0). The QEC magnitude RAM 850applies frequency dependent corrections$k_{{q1},{Heq}} = {{\left( {1 + k_{{i1},{Heq}}} \right)\frac{\left( {1 + k_{i1}} \right)}{\left( {1 + k_{q1}} \right)}} - 1.}$

By applying both LO feed-through compensation and quadrature channelbalancing simultaneously, the mixer output can be described by the moredesirable model:

x _(Mixer,No Imbalances)(t)=A ₀ exp j(ω₀ t+Φ(t)+θ_(i1,eq)+φ₀).

Consequently, the imbalances due to a non-ideal SSB mixer have beencompensated by corrections within the QDWS.

A procedure for determining the exact required V_(L,I), V_(L,Q),H_(eq,I)(f), and H_(eq,Q)(f) based on observable and measurableparameters can still be considered problematic. A solution to such anissue can be facilitated by procedures and techniques that allowindependent measurement of errors due to LO feed-through and quadraturechannel imbalance.

Separating Problem Signals from the Desired Signal

Techniques are herein presented to separate the desired signal from theundesired problem signals in a manner that allows spectral filtering toselect signal components of interest. As in the previous section, onecan begin with techniques for mitigating LO feed-through and follow withI/Q imbalance.

Separating LO Feed-through in Frequency

A technique is now presented herein that separates desired signalcomponents from undesired LO feed-through signals.

Conventional techniques tend to treat QDWS output signals as occupying asignificant majority of the spectrum between −f_(s)/2 and f_(s)/2. Thisis illustrated graphically in Step A of FIG. 9. The baseband signal isthen filtered and applied to the SSB mixer as shown at Step B. Theproblem now is that the LO feed-through is equivalent to a DC signaladded to QDWS output. As such, it exists in the center of the outputsignal spectrum as shown in Step C.

Notch-filtering the LO feed-through at the output of the SSB mixer wouldfilter a portion of the desired signal, thereby degrading the desiredsignal, and ultimately the radar data collected by the receiver. Thisprocess and result is generally undesirable. A frequency bias, however,can be added to the desired signal during its generation by forcing itsspectrum to lie between 0 and f_(s), as is illustrated at Step A in FIG.10. This would allow frequency separation between the desired spectrumand the LO feed-through equivalent, thereby offering an opportunity tofilter the problematic signal without adversely affecting the desiredsignal. A principal difficulty with this approach, however, is that thereconstruction filter as proposed in FIG. 10 is one-sided, that is, itexists for positive frequencies only as shown at Step C, andconsequently is unrealizable. This results in a slightly more elaboratescheme being required.

FIG. 11 depicts an improved scheme for separating LO feed-through fromthe desired signal. In this scheme, all filters are real-valued. Asignal is generated with its spectrum between 0 and f_(s). It isinitially filtered by a realizable reconstruction filter 1110 and thenmodulated upwards in frequency to some intermediate carrier frequency.The extra sideband can be filtered using a sideband filter 1120, leavingthe desired sideband. This is considered by the inventors to be a morecomplicated procedure.

A remaining concern is whether the sideband filter 1120 can suitablyreject the LO feed-through and lower-sideband duplicate chirp.Nevertheless, spectral separation of the desired signal from the LOfeed-through can be achieved.

Another problem with this approach stems from the model presumed for theQDWS output spectrum. As is well known, sampled signals have theirspectrum perfectly replicated around integer multiples of the samplingfrequency f_(s), over all frequencies. Perfect replication over allfrequencies, however, presumes that a train of ideal impulses each witharea proportionate to the instantaneous signal value describes thesamples. Mathematically this is described by:${x_{QDWS}(t)} = {\sum\limits_{n}{{x_{{QDWS},{ideal}}(t)}{\delta \left( {t - {n/f_{s}}} \right)}}}$

where x_(QDWS,ideal)(t) is the desired signal to be reconstructed, andδ(t) is a unit-area impulse function.

Real Digital to Analog Converter (DAC) components, however, do notgenerate ideal impulses, but rather generate voltage values that areheld until the next sample is clocked in. This constitutes an implicitfiltering operation. Mathematically this is described more accurately bythe convolution:${x_{QDWS}(t)} = {\left( {\sum\limits_{n}{{x_{{QDWS},{ideal}}(t)}{\delta \left( {t - {n/f_{s}}} \right)}}} \right)*{{{rect}\left( {f_{s}t} \right)}.}}$

This, of course, implies that the spectrum is weighted (multiplied) by asinc function of the form:${{Spectral}\quad {weighting}} = {\frac{1}{f_{s}}\sin \quad {c\left( \frac{f}{f_{s}} \right)}}$

where:${{sinc}(f)} = {\frac{\sin \quad \left( {\pi \quad f} \right)}{\pi \quad f}.}$

FIG. 12 illustrates how the sinc weighting might affect QDWS outputs.Note that for f=0.9 f_(s) the signal attenuation by the implicitfiltering is almost 20 dB. The remedy for this would be to equalize whatamounts to an effective sinc function filter shape that is applied tothe QDWS output. In a manner similar to the QECDWS scheme hereindescribed, amplitude pre-distortions vs. frequency might be made duringwaveform synthesis such that the desired portion of the reconstructedoutput of the QDWS exhibits lesser (and preferably no) rolloff withfrequency. Some rolloff can in fact be tolerated in LFM chirp waveforms,as downstream components, especially in the multiplier circuits, whichwill tend to flatten the response anyway.

Separating LO Feed-through in Doppler

The detailed description up to now has presumed that one wants tocorrect the signal for each and every pulse, independent of otherpulses. Radar systems, however, typically process multiple pulses toarrive at a final signal-processing product. For example, a typicalhigh-performance Synthetic Aperture Radar (SAR) system might processseveral tens of thousands of pulses to form a single image. Combiningreturns from multiple pulses in a coherent fashion is termed Dopplerprocessing. An adequate presumption for subsequent discussion is that aDoppler frequency shift manifests itself primarily as a pulse-to-pulsephase shift of a signal. The Doppler characteristics of a radar echo canbe manipulated by artificially applying pulse-to-pulse phase shifts insome desirable manner. The LO feed-through can be separated from thedesired signal in its Doppler characteristics.

Consider a receiver video signal model exhibiting only LO feed-throughand no I/Q imbalance, that is:${x_{V}(t)} = \quad \left\lbrack {\left. \begin{matrix}{\exp \quad {j\left( {{M\quad {\omega_{0}\left( {t_{m} - t_{s}} \right)}} + {M\left( {{\Phi \left( {t - t_{s}} \right)} - {\Phi \left( {t - t_{m}} \right)}} \right)} + {M\left( {\phi_{0,s} - \phi_{0,m}} \right)}} \right)} \times} \\\begin{pmatrix}{1 + {{MA}_{L,m}\exp \quad {j\left( {{\Phi \left( {t - t_{m}} \right)} + \theta_{{i1},m} - \theta_{{g0},m} - \phi_{L,m}} \right)}} +} \\{{{MA}_{L,s}\exp} - {j\left( {{\Phi \left( {t - t_{s}} \right)} + \theta_{{i1},s} - \theta_{{g0},s} - \phi_{L,s}} \right)}}\end{pmatrix}\end{matrix} \right\rbrack.} \right.$

A pulse-to-pulse phase shift of π radians can now be added per pulse,such that Φ(t)→(Φ(t)+nπ) is replaced where π is the pulse number index.This effectively alters the reference phase between values of 0 and π ona pulse-to-pulse basis. By noting that exp jnπ=exp−jnπ the video signalthen becomes: ${x_{V}(t)} = \quad {\begin{bmatrix}{\exp \quad {j\left( {{M\quad {\omega_{0}\left( {t_{m} - t_{s}} \right)}} + {M\left( {{\Phi \left( {t - t_{s}} \right)} - {\Phi \left( {t - t_{m}} \right)}} \right)} + {M\left( {\phi_{0,s} - \phi_{0,m}} \right)}} \right)} \times} \\\left( {1 + {\exp \quad {j\left( {n\quad \pi} \right)}\begin{pmatrix}{{{MA}_{L,m}\exp \quad {j\left( {{\Phi \left( {t - t_{m}} \right)} + \theta_{{i1},m} - \theta_{{g0},m} - \phi_{L,m}} \right)}} +} \\{{{MA}_{L,s}\exp} - {j\left( {{\Phi \left( {t - t_{s}} \right)} + \theta_{{i1},s} - \theta_{{g0},s} - \phi_{L,s}} \right)}}\end{pmatrix}}} \right)\end{bmatrix}.}$

The result is that the LO feed-through term is modulated by exp jnπ, butthe desired energy is not. That is, the problematic LO feed-throughsignal has been separated in Doppler from the desired signal.

FIG. 13 illustrates how, if the desired signal 1310 is Dopplerband-limited and centered around a zero Doppler frequency, and if theradar Pulse Repetition Frequency (PRF) 1320 is defined as f_(p), thenthe problematic LO feed-through signal will also be Doppler band-limitedbut will instead be centered around f_(p)/2. Doppler filtering can thenquite effectively mitigate the unwanted energy.

The PRF should be high enough so that the LO feed-through Dopplerspectrum 1330 does not overlap that portion of the desired signalDoppler spectrum of interest. If the Doppler spectrum is limited to abandwidth B_(Doppler), then it can be guaranteed by insistingf_(p)≧2B_(Doppler).

Referring to FIG. 14, the requisite 0/π phase modulation can beaccomplished by inserting the appropriate 0 or π radian phase shift intothe output of the QDWS 1410, prior to the DACs 1420, while the signal isstill just data. It should be noted that a π radian phase shift is justmultiplication by −1. Consequently, exp j(nπ)=(−1)^(n).

An important point that should not be lost is that a particular index nvalue is for a transmitted (TX) pulse and demodulating receiver (RX)pulse pair. Any pulse formed to assist RX demodulation needs to remaincoherent with (i.e. the same pulse-to-pulse phase relationship to) thecorresponding TX signal. That is, n increments on TX only, and remainsunchanged for any RX pulses generated.

Separating I/Q Imbalance Energy

A separating technique can be used to also mitigate I/Q imbalance.Simple 0/π modulation, however, is inadequate for this task.Nevertheless, the principle that appropriate phase shifts in the QDWS ona pulse-to-pulse basis can separate problem signals from the desiredsignal in Doppler is still sound.

Consider again the video signal, but now including the effects of I/Qimbalance as well as LO feed-through: ${x_{V}(t)} = {\begin{bmatrix}{\exp \quad {j\left( {{M\quad {\omega_{0}\left( {t_{m} - t_{s}} \right)}} + {M\left( {{\Phi \left( {t - t_{s}} \right)} - {\Phi \left( {t - t_{m}} \right)}} \right)} + {M\left( {\phi_{0,s} - \phi_{0,m}} \right)}} \right)} \times} \\\begin{pmatrix}{1 + {{MA}_{l,m}\exp \quad j\quad \left( {{2\quad {\Phi \left( {t - t_{m}} \right)}} + {2\quad \theta_{{i1},m}} + \phi_{l,m}} \right)} +} \\{{{MA}_{l,s}\exp} - {j\quad \left( {{2\quad {\Phi \left( {t - t_{s}} \right)}} + {2\quad \theta_{{i1},s}} + \phi_{l,s}} \right)} +} \\{{{MA}_{L,m}\exp \quad {j\left( {{\Phi \left( {t - t_{m}} \right)} + \theta_{{i1},m} - \theta_{{g0},m} - \phi_{L,m}} \right)}} +} \\{{{MA}_{L,s}\exp} - {j\quad \left( {{\Phi \left( {t - t_{s}} \right)} + \theta_{{i1},s} - \theta_{{g0},s} - \phi_{L,s}} \right)}}\end{pmatrix}\end{bmatrix}.}$

Now add a pulse-to-pulse phase shift of some fixed amount, such that(Φ(t)→(Φ(t)+nΔφ) is replaced where Δ100 is the pulse-to-pulse phaseshift, and n is the pulse index number (incrementing by one with eachnew transmitted pulse). Consequently, x_(QDWS)(t) contains a rollingphase shift that rolls at a rate Δφ radians per pulse. The video signalthen becomes: ${x_{V}(t)} = {\begin{bmatrix}{\exp \quad {j\left( {{M\quad {\omega_{0}\left( {t_{m} - t_{s}} \right)}} + {M\left( {{\Phi \left( {t - t_{s}} \right)} - {\Phi \left( {t - t_{m}} \right)}} \right)} + {M\left( {\phi_{0,s} - \phi_{0,m}} \right)}} \right)} \times} \\\begin{pmatrix}{1 + {\exp \quad {{j\left( {2n\quad \Delta \quad \varphi} \right)}\left\lbrack {{MA}_{l,m}\exp \quad j\quad \left( {{2{\Phi \left( {t - t_{m}} \right)}} + {2\theta_{{i1},m}} + \phi_{l,m}} \right)} \right\rbrack}} +} \\{\exp - {{j\left( {2n\quad {\Delta\varphi}} \right)}\left\lbrack {{{MA}_{l,s}\exp} - \quad {j\left( {{2{\Phi \left( {t - t_{s}} \right)}} + {2\theta_{{i1},s}} + \phi_{l,s}} \right)}} \right\rbrack} +} \\{{\exp \quad {{j\left( {n\quad \Delta \quad \varphi} \right)}\left\lbrack {{MA}_{L,m}\exp \quad j\quad \left( {{\Phi \left( {t - t_{m}} \right)} + \theta_{{i1},m} - \theta_{{g0},m} - \phi_{L,m}} \right)} \right\rbrack}} +} \\{\exp \quad - {{j\left( {n\quad \Delta \quad \varphi} \right)}\left\lbrack {{{MA}_{L,s}\exp}\quad - {j\quad \left( {{\Phi \left( {t - t_{s}} \right)} + \theta_{{i1},s} - \theta_{{g0},s} - \phi_{L,s}} \right)}} \right\rbrack}}\end{pmatrix}\end{bmatrix}.}$

A notional video Doppler spectrum is illustrated in FIG. 15. As with the0/π modulation described earlier, problem terms are modulated inDoppler, whereas the desired signal 1510 remains unaffected. Note alsothat the LO feed-through spectrum 1530 and the I/Q imbalance spectrum1520 are separated from each other.

A single point scatterer would form a range-Doppler energy distribution(i.e. SAR image) as illustrated in FIG. 16. Note how the LO feed-throughenergy and I/Q imbalance energy in a focussed SAR image manifestthemselves as a chirp spectrum appropriately offset in Doppler from thedesired impulse 1610 at the image center. While the present inventorspropose that this separation can be exploited as is, they also havenoted that this separation can assist in making independent measurementsof the various undesired signal components to facilitate compensation asdescribed earlier. Nevertheless, if the desired signal is really a setof signals that span a Doppler bandwidth of B_(Doppler), then the LOfeed-through and I/Q imbalance signals will each also contain Dopplerbandwidths of B_(Doppler), but centered at their respective Doppleroffsets. If no contamination of the entire Doppler bandwidth associatedwith the desired signal is insisted on, then a minimum imposed value onthe pulse-to-pulse phase shift will be:${\Delta\varphi} \geq {2{{\pi \left( \frac{B_{Doppler}}{f_{p}} \right)}.}}$

Furthermore, to keep the Doppler bandwidths of aliased undesired signalsfrom contaminating any part of the desired signal's Doppler spectrum,require:${\Delta\varphi} \leq {{\pi \left( {1 - \frac{B_{Doppler}}{f_{p}}} \right)}.}$

These combine to form the restriction f_(p)≧3B_(Doppler).

If, however, one is ultimately interested in merely a lesser portion ofthe desired signal's available Doppler spectrum, then some encroachmentcan be tolerated, and the restriction f_(p)≧3B_(Doppler) can be relaxed.

FIG. 17 illustrates a system 1700 wherein Doppler modulation can beimplemented within a QDWS (ignoring any amplitude corrections). Aparticularly interesting configuration is for Δφ=π/2, which requiresf_(p)≧4B_(Doppler). The following table describes the QDWS outputs forthis case (neglecting amplitude modulations).

QDWS outputs for Δφ=π/2:

(ηΔΦ) ^(χ)QDWS, ι^((t)) ^(χ)QDWS, Q^((t)) 0 cosΦ(t) sinΦ(t) π/2 -sinΦ(t)cosΦ(t) π -cosΦ(t) -sinΦ(t) 3π/2 sinΦ(t) -cosΦ(t)

As indicated, the phase shifts can be implemented as merelytranspositions and negations of the I and Q channel outputs. This mightbe accomplished with suitable switch combinations as illustrated for theQDWS 1810 illustrated in FIG. 18.

In any case, employing rolling phase shifts on a pulse-to-pulse basisallows Doppler separation in the receiver of the desired signal fromthose undesired and due to LO feed-through and quadrature imbalance.Doppler filtering can then mitigate the effects of LO feed-through andquadrature imbalance, without the need for any calibration in thetransmitter.

C. Parameter Estimation for Equalization

Some occasions might nevertheless exist where Doppler filtering is notan available option, perhaps because an adequate radar PRF cannot beemployed. In this situation, balancing and equalization as previouslydescribed are still available to facilitate clean signals. Nocalibration scheme, however, has yet been identified to accomplishbalance and equalization.

The present inventors provide estimation, with some precision, of theparametric values, of LO feed-through and I/Q imbalance (or theirequivalent) using measurements of the video signal. Values so derivedcould then be used to compensate the LO feed-through and equalize theI/Q imbalance in a manner previously discussed. A system can be placedin a calibration or test mode to facilitate these measurements andtechniques, and that many pulses can be integrated to achieve goodSignal to Noise ratios. A test signal can also be employed with adelay-line to avoid any Doppler spreading of the echo signal.

To unambiguously measure a parameter, it must first be isolated fromother obscuring or otherwise confusing factors. To separate the desiredsignal from the LO feed-through, and these from the I/Q imbalanceenergy, Doppler separation can be employed as previously discussed. Thatis, QDWS outputs employing a rolling pulse-to-pulse phase shift can beused. The video signal model then becomes:${x_{V}(t)} = {\begin{bmatrix}{\exp \quad {j\left( {{M\quad {\omega_{0}\left( {t_{m} - t_{s}} \right)}} + {M\left( {{\Phi \left( {t - t_{s}} \right)} - {\Phi \left( {t - t_{m}} \right)}} \right)} + {M\left( {\phi_{0,s} - \phi_{0,m}} \right)}} \right)}} \\{\times \begin{pmatrix}{1 +} \\{{\exp \quad {{j\left( {2n\quad {\Delta\varphi}} \right)}\left\lbrack {{MA}_{I,m}\exp \quad {j\left( {{2{\Phi \left( {t - t_{m}} \right)}} + {2\theta_{{i1},m}} + \phi_{I,m}} \right)}} \right\rbrack}} +} \\{\exp - \quad {{j\left( {2n\quad {\Delta\varphi}} \right)}\left\lbrack {{{MA}_{I,s}\exp} - \quad {j\left( {{2{\Phi \left( {t - t_{s}} \right)}} + {2\theta_{{i1},s}} + \phi_{I,s}} \right)}} \right\rbrack} +} \\{{\exp \quad {{j\left( {n\quad {\Delta\varphi}} \right)}\left\lbrack {{MA}_{L,m}\exp \quad {j\left( {{\Phi \left( {t - t_{m}} \right)} + \theta_{{i1},m} - \theta_{{g0},m} - \phi_{L,m}} \right)}} \right\rbrack}} +} \\{\exp \quad - {{j\left( {n\quad {\Delta\varphi}} \right)}\left\lbrack {{{MA}_{L,s}\exp} - \quad {j\left( {{\Phi \left( {t - t_{s}} \right)} + \theta_{{i1},s} - \theta_{{g0},s} - \phi_{L,s}} \right)}} \right\rbrack}}\end{pmatrix}}\end{bmatrix}.}$

A pulse-to-pulse phase shift can be chosen to allow clear separation andidentification of the LO feed-through from the I/Q imbalance energy. Areasonable value might be Δφ=π/4. This can cause LO feed-through energyto be centered at Doppler frequencies ±f_(p)/8, and I/Q imbalance energyto be centered at Doppler frequencies ±f_(p)/4, thereby separating themfrom the desired signal as well as any aliased energy.

Recall that the following was earlier defined:${{{A_{I}\exp} - {j\quad \phi_{I}}} = {{- \frac{1}{2}}\left( \frac{{\Delta \quad k} - {j\quad \Delta \quad \theta}}{1 + k_{i1} + {j\quad \theta_{q0}}} \right)}},\quad {and}$${A_{L}\exp \quad j\quad \phi_{L}} = {\left( \frac{k_{g0}}{1 + k_{n} + {j\quad \theta_{q0}}} \right).}$

where Δk, Δθ, and k_(i1) are frequency dependent. That is, ifΦ(t)=ω_(T)t, then these depend on ω_(T). Note that θ_(q0) in the presentmodel is a phase shift of a constant LO frequency signal, andconsequently is not normally expected to depend on ω_(T), but yet might.

LO Feed-through Parameter Estimation

With appropriate Doppler processing, a Doppler component exp j(nΔφ) canbe extracted uniquely due to the LO feed-through signal, and can beidentified as: ${{x_{V}(t)}}_{\Delta\varphi} = {{\begin{bmatrix}{\exp \quad {j\left( {{M\quad {\omega_{0}\left( {t_{m} - t_{s}} \right)}} + {M\left( {{\Phi \left( {t - t_{s}} \right)} - {\Phi \left( {t - t_{m}} \right)}} \right)} + {M\left( {\phi_{0,s} - \phi_{0,m}} \right)}} \right)} \times} \\{{MA}_{L,m}\exp \quad {j\left( {{\Phi \left( {t - t_{m}} \right)} + \theta_{{i1},m} - \theta_{{g0},m} - \phi_{L,m}} \right)}}\end{bmatrix}.}}$

A convenient QDWS modulation signal can be chosen (e.g., a sinusoid)where:

Φ(t)=ω_(T) t=2πf _(T) t,

where f_(T) is less than the video cutoff frequency, that is|f_(T)|<B_(V).

Then a video signal can be expanded and modeled by:${{x_{V}(t)}}_{\Delta\varphi} = {{MA}_{L,m}\exp \quad {{j\begin{pmatrix}{{2\pi \quad {f_{T}\left( {t - t_{m}} \right)}} + {{M\left( {\omega_{0} + {2\pi \quad f_{T}}} \right)}\left( {t_{m} - t_{s}} \right)} +} \\{{M\left( {\phi_{0,s} - \phi_{0,m}} \right)} + \theta_{{i1},m} - \theta_{{g0},m} - \phi_{L,m}}\end{pmatrix}}.}}$

Performing a Fourier Transform over the envelope finite interval−T/2≦(t−t_(m))≦T/2 yields:${{X_{V}(f)}}_{\Delta\varphi} = {{TMA}_{L,m}\exp \quad {j\begin{pmatrix}{{{M\left( {\omega_{0} + {2\pi \quad f_{T}}} \right)}\left( {t_{m} - t_{s}} \right)} +} \\{\left( {\phi_{0,s} - \phi_{0,m}} \right) + \theta_{{i1},m} - \theta_{{g0},m} - \phi_{L,m}}\end{pmatrix}}{{{sinc}\left( \frac{f - f_{T}}{T} \right)}.}}$

In particular, at the spectral peak, we identify:${{X_{V}\left( f_{T} \right)}}_{\Delta\varphi} = {{TMA}_{L,m}\exp \quad {j\begin{pmatrix}{{{M\left( {\omega_{0} + {2\pi \quad f_{T}}} \right)}\left( {t_{m} - t_{s}} \right)} +} \\{{M\left( {\phi_{0,s} - \phi_{0,m}} \right)} + \theta_{{i1},m} - \theta_{{g0},m} - \phi_{L,m}}\end{pmatrix}}}$

which has magnitude:${{{X_{V}\left( f_{T} \right)}}_{\Delta\varphi}} = {{TMA}_{L,m} = {\frac{{TMk}_{g0}}{\sqrt{\left( {1 + k_{i1}} \right)^{2} + \left( \theta_{q0} \right)^{2}}}.}}$

The known parameters are T, M, ω₀, f_(T), t_(m), t_(s), and of courseX_(V)(f_(T)). The unknown parameters are k_(g0), k_(i1), θ_(i1), θ_(q0),and θ_(g0). The best we can do here is ascertain the quantitiesk_(g0)/√{square root over ((1+k_(i1))²+(θ_(q0))²)}{square root over((1+k_(i1))²+(θ_(q0))²)} and└M(φ_(0,s)−φ_(0,m))+θ_(i1,m)−θ_(g0,m)−φ_(L,m)┘ which still leaves someambiguity in the specific parameters whose value is desired to be known.Identifying precisely k_(g0), and θ_(g0) is problematic given that theyare perturbed by unknown values of k_(i1) and θ_(i1), among others.

Rather than identify these parameters specifically, it should berecognized given the frequency that the aim of calibration is toascertain the DC offsets to minimize the offending components in thereceived signal.

Equivalent SSB Mixer DC Offset Estimation

Reexamining the earlier discussion about compensating LO feed-throughwith a DC offset into the SSB mixer, it should be recognized that themixer output with no QDWS input signal is given by (ignoring I/Qimbalance and LO phase shift term θ_(q0) for the moment):${x_{Mixer}(t)} = {\frac{1}{2}\left( {{k_{g0}{\cos \left( \theta_{g0} \right)}} + {{jk}_{g0}{\sin \left( \theta_{g0} \right)}}} \right)\exp \quad {j\left( {\omega_{0}t} \right)}}$

with magnitude k_(g0)/2. Even small values for θ_(q0) won't change thismuch.

On the other hand, with DC offsets applied to the SSB mixer input, aspreviously discussed, the output is described by:${x_{Mixer}(t)} = {\frac{1}{2}\begin{pmatrix}{\left\lfloor {{V_{L,I}{H_{I}(0)}} + {k_{g0}{\cos \left( \theta_{g0} \right)}} - {V_{L,Q}{H_{Q}(0)}{\sin \left( \theta_{q0} \right)}}} \right\rfloor +} \\{j\left\lbrack {{V_{L,Q}{H_{Q}(0)}{\cos \left( \theta_{q0} \right)}} + {k_{g0}{\sin \left( \theta_{g0} \right)}}} \right\rbrack}\end{pmatrix}\exp \quad {{j\left( {\omega_{0}t} \right)}.}}$

By now defining:${{k_{{g0},{eq}} = \sqrt{\left( {{V_{L,I}{H_{I}(0)}} + {k_{g0}{\cos \left( \theta_{g0} \right)}} - {V_{L,Q}{H_{Q}(0)}{\sin \left( \theta_{q0} \right)}}} \right)^{2} + \left( {{V_{L,Q}{H_{Q}(0)}{\cos \left( \theta_{q0} \right)}} + {k_{g0}{\sin \left( \theta_{g0} \right)}}} \right)^{2}}},{and}}\quad$$\quad {A_{L,m,{eq}} = \frac{k_{{g0},{eq}}}{\sqrt{\left( {1 + k_{i1}} \right)^{2} + \left( \theta_{q0} \right)^{2}}}}$

the measured video signal spectral peak, when a DC offset is applied tothe SSB mixer input, becomes:${{{X_{V}\left( f_{T} \right)}}_{\Delta\varphi}} = {{TMA}_{L,m,{eq}} = {\frac{{TMk}_{{g0},{eq}}}{\sqrt{\left( {1 + k_{i1}} \right)^{2} + \left( \theta_{q0} \right)^{2}}}.}}$

The task at hand is now to find V_(L,I) and V_(L,Q) to force as nearlyas possible k_(g0,eq)=0, and thereby X_(V)(f_(T))|_(Δφ)=0. Clearly,these DC offsets also encompass the effects of other relevant factorsinfluencing LO feed-through, such as H_(I)(0) and H_(Q)(0). In any case,the essential task is to find appropriate parameters V_(L,I) and V_(L,Q)to nullify LO feed-through.

Of interest should be the fact that if θ_(q0)=0, then V_(L,I) andV_(L,Q) can be independently minimized to achieve an overall solution.That is, the value for V_(L,I) that minimizes k_(g0,eq) does soregardless of V_(L,Q), and likewise the value for V_(L,Q) that minimizesk_(g0,eq) does so regardless of V_(L,I). However, generally θq0≠0, sotheir values do influence each other somewhat, although probably notvery much since θ_(q0) is typically expected to be small.

Several possible techniques to find optimal V_(L,I) and V_(L,Q) arepresented herein.

Technique 1

The first procedure amounts to guessing about new solutions and keepingthe better solution between the new and the old.

1) Guess an initial value for V_(L,I) and V_(L,Q), and calculate|X_(V)(f_(T))|_(Δφ)|.

2) Guess another value for V_(L,I) and recalculate |X_(V)(f_(T))|_(Δφ)|.

3) Remember the value for V_(L,I) that gave the minimum|X_(V)(f_(T))|_(Δφ)|.

4) Guess another value for V_(L,Q) and recalculate |X_(V)(f_(T))|_(Δφ)|.

5) Remember the value for V_(L,Q) that gave the minimum|X_(V)(f_(T))|_(Δφ)|.

6) Return to step 2 and repeat until performance requirements are met.

Technique 2—Systematic Search

The procedure becomes to alternately ‘tune’ each of V_(L,I) and V_(L,Q)to a minimum |X_(V)(f_(T))|_(Δφ)|.

1) Set initial values for V_(L,I) and V_(L,Q).

2) Begin with V_(L,I) at its absolute minimum value and calculate|X_(V)(f_(T))|_(Δφ)|.

3) Set an initial step size for V_(L,I) at some positive ΔV.

4) Increment V_(L,I) by ΔV and recalculate |X_(V)(f_(T))|_(Δφ)|.

5) Repeat step 4 until the new |X_(V)(f_(T))|_(Δφ)| is greater than theold.

6) Adjust the step size and direction such that ΔV_(new)=−ΔV_(old)/2.

7) Go back to step 4 and repeat until no discernable improvement isobserved.

8) Repeat steps 2 through 7 for V_(L,Q).

9) Repeat steps 2 through 7 again for V_(L,I).

10) Repeat steps 8 and 9 until performance criteria are met.

Technique 3—Gradient Search

The equation of the energy at f_(T) is:${{{X_{V}\left( f_{T} \right)}}_{\Delta\varphi}}^{2} = {\left\lbrack \frac{({TM})^{2}}{\left( {1 + k_{i1}} \right)^{2} + \left( \theta_{q0} \right)^{2}} \right\rbrack \left( k_{{g0},{eq}} \right)^{2}}$

where k_(g0,eq) depends on both V_(L,I) and V_(L,Q). Plotting energy asa function V_(L,I) and V_(L,Q) shows an error surface or bowl over theV_(L,I), V_(L,Q) plane. This error surface has both its local and itsglobal minimum at the previously stated values:${V_{L,Q} = {- \left( \frac{k_{g0}{\sin \left( \theta_{g0} \right)}}{{H_{Q}(0)}{\cos \left( \theta_{q0} \right)}} \right)}},\quad {and}$$V_{L,I} = {- {\left( \frac{{k_{g0}{\cos \left( \theta_{g0} \right)}} - {V_{L,Q}{H_{Q}(0)}{\sin \left( \theta_{q0} \right)}}}{H_{i}(0)} \right).}}$

Note also that the minimum value for V_(L,I) depends on the currentvalue for V_(L,Q) if θ_(q0)≠0. Additionally, the greater the errors inV_(L,I) and V_(L,Q), the steeper is the gradient of the error surface.This leads to the techniques of gradient searches to find optimalV_(L,I) and V_(L,Q). Specifically, we implement:${{{{{V_{L,Q,{new}} = {V_{L,Q,{old}} - {\mu \frac{}{V_{L,Q}}{{X_{V}\left( f_{T} \right)}}_{\Delta\varphi}}}}}^{2},\quad {and}}{V_{L,I,{new}} = {V_{L,I,{old}} - {\mu \frac{}{V_{L,I}}{{X_{V}\left( f_{T} \right)}}_{\Delta\varphi}}}}}}^{2},$

where μ is a convergence constant, usually chosen to be small enough toallow gradual convergence.

Derivatives can be estimated by making measurements of the energy atf_(T) for slightly different values of offset voltage, that is:${{{{{{\frac{}{V_{L,Q}}{{X_{V}\left( f_{T} \right)}}_{\Delta\varphi}}}^{2} \approx \frac{{{\Delta {{X_{V}\left( f_{T} \right)}}_{\Delta\varphi}}}^{2}}{\Delta \quad V_{L,Q}}},\quad {and}}{\frac{}{V_{L,I}}{{X_{V}\left( f_{T} \right)}}_{\Delta\varphi}}}}^{2} \approx {\frac{{{\Delta {{X_{V}\left( f_{T} \right)}}_{\Delta\varphi}}}^{2}}{\Delta \quad V_{L,I}}.}$

A suitable algorithm can then be:

1) Select initial V_(L,I) and V_(L,Q).

2) Measure the energy at f_(T) for V_(L,Q)−(ΔV_(L,Q)/2) andV_(L,Q)+(ΔV_(L,Q)/2), and estimate error gradient.

3) Update estimate for V_(L,Q).

4) Measure the energy at f_(T) for V_(L,I)−(ΔV_(L,I)/2) andV_(L,I)+(ΔV_(L,I)/2), and estimate error gradient.

5) Update estimates for V_(L,I).

6) Go back to step 2 and repeat until derivative estimates are zero (orsuitably small), or some other exit criteria is met.

The inventors in a simulation implemented the procedure with resultsdisplayed in the graph illustrated in FIG. 19.

The nature of this gradient'search technique is that the final imbalancelevel will be limited to somewhat below the noise floor of the desiredsignal, depending on convergence parameter μ. In general, smaller valuesfor μ would allow convergence to a lower noise floor, but cause slowerconvergence. Schemes might be employed whereby μ adapts from largervalues to smaller values either with time, or as residual imbalanceenergy diminishes. This kind of adaptive algorithm sometimes goes by thename “simulated annealing.”

Furthermore, the Fourier Transform, while useful in enhancing SNR, isn'tcompletely necessary for determining the presence of LO feed-throughenergy for a frequency independent leakage. The leakage (subject tonoise) is generally time independent as well, and exists for each timesample. That is:${{{{{{x_{V}(t)}}_{\Delta\varphi}} = {{x_{V}\left( t_{m} \right)}}_{\Delta\varphi}}} = \left( \frac{{Mk}_{{g0},{eq}}}{1 + k_{n}} \right)},$

which implies that any one sample of time domain video signal isadequate to indicate relative magnitudes for k_(g0,eq). Consequently,one could generally avoid the Fourier Transform and substitute into thealgorithms a scaled time sample of the video signal:

|X _(V)(f _(T))|_(Δφ) |→T|x _(V)(t _(m))|_(Δφ)|.

A somewhat in-between substitution that works for reasonably good videosignal SNR, that still doesn't involve a Fourier Transform, is thesubstitution of a non-coherent magnitude integration over time, namely:

|X _(V)(f _(T))|_(Δφ) |→∫|x _(V)(t)|_(Δφ) |dt.

If SNR is still problematic, more pulses might be coherently combined inDoppler processing, prior to any video signal component analysis.

Frequency-dependent LO Feed-through Parameter Estimation

The previous model presumes that V_(L,I) and V_(L,Q) are true constants,independent of frequency f_(T). In the event that some dependence onf_(T) does in fact exist, then the aforementioned procedures would needto be repeated for other sample modulation frequencies f_(T). V_(L,I)and V_(L,Q) values would be determined that depend on an instantaneousfrequency indication from the QDWS, and then could be interpolated forfrequencies in-between those in our calibration sample set.

With the QDWS offering only sinusoidal signals at constant frequencyf_(T), the video signal component of interest contains spectral energyat f_(T). If f_(T) is outside of the video passband, then it can beseverely attenuated by the video filter, which is of course preciselywhat a video filter is suppose to do. Whatever attenuated energy doespass through the video filter, if beyond the Nyquist frequency in asampled signal, can be aliased to an equivalent in-band frequency. Aslong as the offending energy is adequately attenuated for QDWS constantfrequency modulation, balancing the QDWS at frequencies |f_(T)|>B_(V) isgenerally of little concern.

For LFM chirp signals, however, where bandwidth substantially greaterthan the video bandwidth is experienced, balancing the QDWS frequencies|f_(T)|>B_(V) becomes important.

Effects of LFM Chirps on LO Feed-through Compensation

To properly understand this topic, the generic video signal componentcan be returned to and with it a LFM chirp can be incorporated bysubstituting Φ(t)=ω_(T)t+(γ_(T)/2)t² where γ_(T) is the chirp rate. Thisyields a Doppler spectral component:${{x_{V}(t)}}_{\Delta\varphi} = {{{MA}_{L,m}\begin{bmatrix}{\exp \quad {j\begin{pmatrix}{{{M\left( {\omega_{0} + \omega_{T}} \right)}\left( {t_{m} - t_{s}} \right)} + {M\frac{\gamma_{T}}{2}\left( {t_{m} - t_{s}} \right)^{2}} +} \\{{M\left( {\phi_{0,s} - \phi_{0,m}} \right)} + \theta_{{i1},m} - \theta_{{g0},m} - \phi_{L,m}}\end{pmatrix}} \times} \\{\exp \quad {j\left( {{\left( {\omega_{T} + {M\quad {\gamma_{T}\left( {t_{m} - t_{s}} \right)}}} \right)\left( {t - t_{m}} \right)} + {\frac{\gamma_{T}}{2}\left( {t - t_{m}} \right)^{2}}} \right)}}\end{bmatrix}}.}$

The QDWS output, appropriately delayed, has instantaneous frequency:$\omega_{\Phi,{inst}} = {{\frac{}{t}{\Phi \left( {t - t_{m}} \right)}} = {\omega_{T} + {\gamma_{T}\left( {t - t_{m}} \right)}}}$

whereas the video signal has instantaneous frequency:

ω_(V,inst)=ω_(T) +Mγ _(T)(t _(m) −t _(s))+γ_(T)(t−t _(m))=ω_(Φ,inst) +Mγ_(T)(t _(m) −t _(s))

In units of Hz, this would be:$f_{V.{inst}} = {f_{\Phi,{inst}} + {M\frac{\gamma_{T}}{2\pi}{\left( {t_{m} - t_{s}} \right).}}}$

This also states that any portion of the QDWS chirp can be translated toany other video frequency range by adjusting the relative delay(t_(m)−t_(s)).

The video filter limits |f_(V,inst)≦B_(V), so that the QDWS frequenciesobservable in the video are:$\left\lbrack {{- B_{V}} - {M\frac{\gamma_{T}}{2\pi}\left( {t_{m} - t_{s}} \right)}} \right\rbrack \leq f_{\Phi,{inst}} \leq {\left\lbrack {B_{V} - {M\frac{\gamma_{T}}{2\pi}\left( {t_{m} - t_{s}} \right)}} \right\rbrack.}$

Clearly, when employing a LFM chirp, one can observe any QDWSinstantaneous frequency that is chosen, using any video filter bandwidthavailable, by simply adjusting the relative delay (t_(m)−t_(s)). Thereare, however, important side effects to bear in mind. Specifically, thetime interval over which can be observed where the problem energy issomewhat less than the entire pulse, which also impacts the total energythat is observable in this component of the video signal. This isillustrated in the time-frequency diagram in FIG. 20.

The total time interval over which the LO feed-through generated chirpis observable (within the video filter passband) is:$T_{obs} = {T\left( \frac{2B_{V}}{B_{\Phi}} \right)}$

where,$B_{\Phi} = {{\frac{\gamma_{T}}{2\pi}T} = {{the}\quad {total}\quad {chirp}\quad {bandwidth}\quad {at}\quad {the}\quad {QDWS}\quad {{output}.}}}$

The observation interval is centered att−t_(m)=−(ω_(T)/γ_(T)+M(t_(m)−t_(s))). One can presume that thisdominates all the other implied envelopes, such that the actual envelopefor the problem chirp can be approximated by this observation interval,that is, the observation envelope is given by:${{rect}\left( \frac{t - t_{m} + \left( {{\omega_{T}/\gamma_{T}} + {M\left( {t_{m} - t_{s}} \right)}} \right)}{T_{obs}} \right)},$

which can now overtly be included in the video signal, describing it as:${{x_{V}(t)}}_{\Delta\varphi} = {{{MA}_{L,m}\begin{bmatrix}{\exp \quad {j\begin{pmatrix}{{{M\left( {\omega_{0} + \omega_{T}} \right)}\left( {t_{m} - t_{s}} \right)} + {M\frac{\gamma_{T}}{2}\left( {t_{m} - t_{s}} \right)^{2}} +} \\{{M\left( {\phi_{0,s} - \phi_{0,m}} \right)} + \theta_{{i1},m} - \theta_{{g0},m} - \phi_{L,m}}\end{pmatrix}} \times} \\{\exp \quad {j\left( {{\left( {\omega_{T} + {M\quad {\gamma_{T}\left( {t_{m} - t_{s}} \right)}}} \right)\left( {t - t_{m}} \right)} + {\frac{\gamma_{T}}{2}\left( {t - t_{m}} \right)^{2}}} \right)} \times} \\{{rect}\left( \frac{t - t_{m} + \left( {{\omega_{T}/\gamma_{T}} + {M\left( {t_{m} - t_{s}} \right)}} \right)}{T_{obs}} \right)}\end{bmatrix}}.}$

Nevertheless, within the observation window, a particular observationtime corresponds directly to a QDWS instantaneous frequency by:$f_{\Phi,{inst}} = {\frac{\omega_{T}}{2\pi} + {\frac{\gamma_{T}}{2\pi}{\left( {t - t_{m}} \right).}}}$

Consequently, mitigating LO feed-through at a particular QDWSinstantaneous output frequency f_(Φ,inst) can be observed by nulling anyenergy at a particular observation time (t−t_(m)), given that theobservation time is in fact observable—that is, within the enveloperect((t−t_(m)+(ω_(T)/γ_(T)+M(t_(m)−t_(s))))/T_(obs)). All of this canlead to employment of generic procedures for identifying optimal V_(L,I)and V_(L,Q) as functions of instantaneous QDWS output frequencyf_(Φ,inst).

Procedure for Frequency-dependent LO Feed-through Equalization

An overall algorithm for identifying frequency dependent values forV_(L,I) and V_(L,Q) can be adopted as follows.

1) Divide the QDWS output frequency range into segments no wider than2B_(V).

2) For each frequency segment, center it in the video passband withappropriate selection of (t_(m)−t_(s)).

3) Generate data for new V_(L,I) and V_(L,Q) functions, and extract theappropriate video component x_(V)(t)|_(Δφ).

4) Update estimates of more optimal V_(L,I) and V_(L,Q) as a function ofappropriate f_(Φ,inst) based on measurements of energy at correspondingtimes (t−t_(m)).

5) Go back to step 3 and repeat until satisfied with the results in thisfrequency range.

6) Go back to step 2 and repeat for the next frequency range until done.

With this procedure, no Fourier Transform is required. This also meansthat no corresponding benefit in noise reduction is available that theFourier Transform brought. Consequently, adequate noise reduction mightrequire enhanced coherent processing gain from Doppler processing, whichis, collecting more pulses. For example, if −60 dB LO feed-throughsuppression is desired, then enough pulses need to be collected toachieve in the neighborhood of 60 dB SNR in the primary signal's Dopplercell.

Any of the techniques previously discussed for updating V_(L,I) andV_(L,Q) (genetic search, systematic searching, or gradient searching)can be used, but with the substitute:$\left. {{{X_{V}\left( f_{T} \right)}}_{\Delta\varphi}}\rightarrow{{{x_{V}\left( {t_{m} + \frac{{2\pi \quad f_{\Phi,{inst}}} - {\omega_{T}}_{\quad}}{\gamma_{T}}} \right)}}_{\Delta\varphi}.} \right.$

This is the result of employing a LFM chirp instead of a constantfrequency.

Frequency Dependent I/Q Imbalance Parameter Estimation

With appropriate Doppler processing a Doppler component exp j(2nΔφ) canbe extracted uniquely due to the I/Q imbalance signal in a mannersimilar to identifying the LO feed-through signal, and identified as:${{x_{V}(t)}}_{2{\Delta\varphi}} = {{{MA}_{I,m}\begin{bmatrix}{\exp \quad {j\left( {{M\quad {\omega_{0}\left( {t_{m} - t_{s}} \right)}} + {M\left( {\phi_{0,s} - \phi_{0,m}} \right)} + {2\theta_{{i1},m}} + \phi_{I,m}} \right)} \times} \\{\exp \quad {j\left( {M\left( {{\Phi \left( {t - t_{s}} \right)} - {\Phi \left( {t - t_{m}} \right)}} \right)} \right)} \times} \\{\exp \quad {j\left( {2{\Phi \left( {t - t_{m}} \right)}} \right)}}\end{bmatrix}}.}$

The phase function Φ(t) will contain time-varying frequencies by design,which with this feature of the invention can be a LFM chirp.Consequently, the entities A_(I,m), θ_(i1,m), and φ_(I,m) can alsodisplay their frequency dependencies in the video signal. Furthermore,as with the notion of a frequency-dependent LO feed-through parameter,quadrature balance can be achieved when we force A_(I,m)=0 over theentire range of QDWS frequencies. This implies, as described in theprevious section, that measurements made with a single frequency will beinadequate to the task. That is, measurements need to be made with:

Φ(t)=ω_(T) T+(γ_(T)/2)t ².

With the LFM chirp, the video signal component of interest becomes:${{x_{V}(t)}}_{2{\Delta\varphi}} = {{{MA}_{I,m}\begin{bmatrix}{\exp \quad {j\begin{pmatrix}{{{M\left( {\omega_{0} + \omega_{T}} \right)}\left( {t_{m} - t_{s}} \right)} + {M\frac{\gamma_{T}}{2}\left( {t_{m} - t_{s}} \right)^{2}} +} \\{{M\left( {\phi_{0,s} - \phi_{0,m}} \right)} + {2\theta_{{i1},m}} + \phi_{I,m}}\end{pmatrix}} \times} \\{\exp \quad {j\left( {{\left( {{2\omega_{T}} + {M\quad {\gamma_{T}\left( {t_{m} - t_{s}} \right)}}} \right)\left( {t - t_{m}} \right)} + {\gamma_{T}\left( {t - t_{m}} \right)}^{2}} \right)}}\end{bmatrix}}.}$

The QDWS output, appropriately delayed, as before, has instantaneousfrequency:${\omega_{\Phi,{inst}} = {{\frac{}{t}{\Phi \left( {t - t_{m}} \right)}} = {\omega_{T} + {\gamma_{T}\left( {t - t_{m}} \right)}}}},$

whereas the video signal has instantaneous frequency:

ω_(V,inst)=ω_(T) +Mγ _(T)(t _(m) −t _(s))+γ_(T)(t−t _(m))=ω_(Φ,inst) +Mγ_(T)(t _(m) −t _(s))

In units of Hz, this would be:$f_{V,{inst}} = {f_{\Phi,{inst}} + {M\frac{\gamma_{T}}{2\pi}{\left( {t_{m} - t_{s}} \right).}}}$

As with the LO feed-through, this process again states that any portionof the QDWS chirp can be translated to any other video frequency rangeby adjusting the relative delay (t_(m)−t_(s)).

The video filter limits |f_(V,inst)|≦B_(V), so that the QDWS frequenciesobservable in the video due to I/Q imbalance become:$\left\lbrack \frac{{- B_{V}} - {M\frac{\gamma_{T}}{2\pi}\left( {t_{m} - t_{s}} \right)}}{2} \right\rbrack \leq f_{\Phi,{inst}} \leq {\left\lbrack \frac{B_{V} - {M\frac{\gamma_{T}}{2\pi}\left( {t_{m} - t_{s}} \right)}}{2} \right\rbrack.}$

Again, when employing a LFM chirp, one can observe any QDWSinstantaneous frequency that is chosen, using any video filter bandwidthavailable, by simply adjusting the relative delay (t_(m)−t_(s)).

FIG. 20 is now shown slightly modified in the illustration of FIG. 21,so that the time interval over which one might observe the problemenergy is still somewhat less than the entire pulse, which again alsoimpacts the total energy that is observable in this component of thevideo signal.

The total time interval over which the LO feed-through generated chirpis observable (within the video filter passband) is now:$T_{obs} = {T\left( \frac{B_{V}}{B_{\Phi}} \right)}$

which is reduced somewhat from the LO feed-through case.

The observation interval is now centered att−t_(m)=−(ω_(T)/γ_(T)+M(t_(m)−t_(s))/2). One can presume that thisdominates all the other implied envelopes, such that the actual envelopefor the problem chirp can be approximated by this observationinterval—that is:${{rect}\left( \frac{t - t_{m} + \left( {{\omega_{T}/\gamma_{T}} + {{M\left( {t_{m} - t_{s}} \right)}/2}} \right)}{T_{obs}} \right)}.$

which can now be overtly included in the video signal, describing it as:${{x_{V}(t)}}_{2{\Delta\varphi}} = {{{MA}_{l,m}\begin{bmatrix}{\exp \quad {j\begin{pmatrix}{{{M\left( {\omega_{0} + \omega_{T}} \right)}\left( {t_{m} - t_{s}} \right)} + {M\frac{\gamma_{T}}{2}\left( {t_{m} - t_{s}} \right)^{2}} +} \\{{M\left( {\phi_{0,s} - \phi_{0,m}} \right)} + {2\theta_{{il},m}} + \phi_{1,m}}\end{pmatrix}} \times} \\{\exp \quad {j\left( {{\left( {{2\omega_{T}} + {M\quad {\gamma_{T}\left( {t_{m} - t_{s}} \right)}}} \right)\left( {t - t_{m}} \right)} + {\gamma_{T}\left( {t - t_{m}} \right)}^{2}} \right)} \times} \\{{rect}\left( \frac{t - t_{m} + \left( {{\omega_{T}/\gamma_{T}} + {{M\left( {t_{m} - t_{s}} \right)}/2}} \right)}{T_{obs}} \right)}\end{bmatrix}}.}$

Nevertheless, within the observation window, a particular observationtime still corresponds directly to a QDWS instantaneous frequency by:$f_{\Phi,{inst}} = {\frac{\omega_{T}}{2\pi} + {\frac{\gamma_{T}}{2\pi}{\left( {t - t_{m}} \right).}}}$

Consequently, mitigating I/Q imbalance at a particular QDWSinstantaneous output frequency f_(Φ,inst) can be observed by nulling anyenergy at a particular observation time (t−t_(m)), given that theobservation time is in fact observable, that is, within the envelope:rect((t−t_(m)+(ω_(T)/γ_(T)+M(t_(m)−t_(s))/2))/T_(obs)).

The procedure calls for the initial selection of a particular value(t_(m)−t_(s)) to thereby select an appropriate part of the QDWS outputspectrum as observable within the video bandwidth. Then a particularQDWS output frequency f_(Φ,inst) is identified as corresponding to theenergy at time (t−t_(m))=(2πf_(Φ,inst)−ω_(T))/γ_(T).

At this particular time the following is identified:${{{x_{V}(t)}{_{2{\Delta\varphi}}}} = {{MA}_{1,m} = {\frac{M}{2}\left( \frac{\sqrt{{\Delta \quad k^{2}} + {\Delta\theta}^{2}}}{\sqrt{\left( {1 + k_{n}} \right)^{2} + \theta_{q0}^{2}}} \right)}}}$

where Δk=(k_(q1,eq)−k_(i1,eq)), and Δθ=(θ_(q1,eq)−θ_(i1,eq)−θ_(q0)), andfurthermore:

k _(i1,eq)=(1+k _(i1,Heq))(1+k _(i1))−1,

 θ_(i1,eq)=θ_(i1)+θ_(i1,Heq),

k _(q1,eq)=(1+k _(q1,Heq))(1+k _(q1))−1, and

θ_(q1,eq)=θ_(q1)+θ_(q1,Heq).

The adjustments available to the equalization process are k_(i1,Heq),θ_(i1,Heq), k_(q1,Heq), and θ_(q1,Heq). Typically,k_(i1,Heq)=θ_(i1,Heq)=0 will be assumed so that at the particular time(t−t_(m))=(2πf_(Φ,inst)−ω_(T))/γ_(T):

Δk=(1+k _(q1,Heq))(1+k _(q1))−1−k _(i1), and

Δθ=_(q1)+θ_(q1,Heq)−θ_(i1)−θ_(q0).

Given these equivalencies and conditions:${{{x_{V}(t)}{_{2{\Delta\varphi}}}^{2}} = {\left( \frac{M}{2} \right)^{2}{\left( \frac{\left( {\left( {{\left( {1 + k_{{q1},{Heq}}} \right)\left( {1 + k_{q1}} \right)} - 1 - k_{i1}} \right)^{2} + \left( {\theta_{q1} + \theta_{{q1},{Heq}} - \theta_{i1} - \theta_{q0}} \right)^{2}} \right)}{\left( {\left( {1 + k_{i1}} \right)^{2} + \theta_{q0}^{2}} \right)} \right).}}}$

That is, finding correct k_(q1,Heq) and θ_(q1,Heq) for QDWS frequencyf_(Φ,inst) will cause an energy null at time(t−t_(m))=(2πf_(Φ,inst)−ω_(T))/γ_(T).

Simply restated, the idea is to adjust k_(q1,Heq) and θ_(q1,Heq) until|x_(V)(t)|_(2Δφ)|²=0 for all f_(Φ,inst).

A variety of techniques might be used to find optimal k_(q1,Heq) andθ_(q1,Heq), including genetic algorithms, systematic search algorithms,and gradient search algorithms (as is the case with the LO feed-throughanalysis). Of note is that unlike V_(L,I) and V_(L,Q) for the LOfeed-through problem, the influences of k_(q1,Heq) and θ_(q1,Heq) on|x_(V)(t)|_(2Δφ)|²=0 are independent of the other. That is, each can beoptimized independent of the other. All of this leads to a genericprocedure for identifying optimal k_(q1,Heq) and θ_(q1,Heq) as functionsof instantaneous QDWS output frequency f_(Φ,inst).

Procedure for Frequency-dependent Parameter Estimation

An overall algorithm and process for identifying frequency dependentvalues for k_(q1,Heq) and θ_(q1,Heq) can be as follows:

1) Divide the QDWS output frequency range into segments no wider thanB_(V).

2) For each frequency segment, center it in the video passband withappropriate selection of (t_(m−t) _(s)).

3) Generate data for new k_(q1,Heq) and θ_(q1,Heq) functions, andextract the appropriate video component X_(V)(t)|_(2Δφ).

4) Update estimates of more optimal k_(q1,Heq) and θ_(q1,Heq) as afunction of appropriate f_(Φ,inst) based on measurements of energy atcorresponding times (t−t_(m)).

5) Go back to step 3 and repeat until satisfied with the results in thisfrequency range.

6) Go back to step 2 and repeat for the next frequency range until done.

As with the similar technique for LO feed-through suppression, noFourier Transform is required. This again means that no correspondingbenefit in noise reduction is available that the Fourier Transformbrought. Consequently, adequate noise reduction might require enhancedcoherent processing gain from Doppler processing, which is, collectingmore pulses.

A variety of techniques can be used for updating k_(q1,Heq) andθ_(q1,Heq) (genetic search, systematic searching, or gradientsearching). The gradient search algorithm is herein described in detail.

Gradient Search for I/Q Balance

Recall that the instantaneous power in the appropriate component of thevideo signal being described by:${{{x_{V}(t)}{_{2{\Delta\varphi}}}^{2}} = {\left( \frac{M}{2} \right)^{2}\left( \frac{\left( {\left( {{\left( {1 + k_{{q1},{Heq}}} \right)\left( {1 + k_{q1}} \right)} - 1 - k_{i1}} \right)^{2} + \left( {\theta_{q1} + \theta_{{q1},{Heq}} - \theta_{i1} - \theta_{q0}} \right)^{2}} \right)}{\left( {\left( {1 + k_{i1}} \right)^{2} + \theta_{q0}^{2}} \right)} \right)}}$

Note that this represents an error surface or bowl over the k_(q1,Heq),θ_(q1,Heq) plane. As previously stated, this error surface has both itslocal and its global minimum at:${k_{{q1},{Heq}} = {{\left( {1 + k_{{i1},{Heq}}} \right)\frac{\left( {1 + k_{i1}} \right)}{\left( {1 + k_{q1}} \right)}} - 1}},\quad {and}$θ_(q1, Heq) = θ_(i1) − θ_(q1) + θ_(i1, Heq) + θ_(q0),

where it can normally be assumed that k_(i1,Heq)=θ_(i1,Heq)=0.

The gradient search algorithm is implemented via the updates:$k_{{q1},{Heq},{new}} = {k_{{q1},{Heq},{old}} - {\mu \frac{\quad}{k_{{q1},{Heq}}}{{{{x_{V}(t)}{_{2{\Delta\varphi}}}^{2}},\quad {{{and}\theta_{{q1},{Heq},{new}}} = {\theta_{{q1},{Heq},{old}} - {\mu \frac{\quad}{\theta_{{q1},{Heq}}}{{{{x_{V}(t)}{_{2{\Delta\varphi}}}^{2}},}}}}}}}}}$

where μ is a convergence constant, usually chosen to be small enough toallow gradual convergence.

The derivatives can be estimated by making measurements of theinstantaneous power at times (t−t_(m)) for slightly different values ofthe parameters in control and which are sought to br optimized, where:$\frac{\quad}{k_{{q1},{Heq}}}{{{{{x_{V}(t)}{_{2{\Delta\varphi}}}^{2}} \approx \frac{\Delta {{{x_{V}(t)}{_{2{\Delta\varphi}}}^{2}}}}{\Delta \quad k_{{q1},{Heq}}}},\quad {{and}\frac{\quad}{\theta_{{q1},{Heq}}}{{{{x_{V}(t)}{_{2{\Delta\varphi}}}^{2}} \approx {\frac{\Delta {{{x_{V}(t)}{_{2{\Delta\varphi}}}^{2}}}}{\Delta \quad \theta_{{q1},{Heq}}}.}}}}}}$

A suitable algorithm and process can then be:

1) Select initial k_(q1,Heq) and θ_(q1,Heq).

2) Measure the energy at (t−t_(m)) for k_(q1,Heq)±(Δk_(q1,Heq)/2).

3) Update estimate for k_(q1,Heq).

4) Measure the energy at (t−t_(m)) for θ_(q1,Heq)±(Δθ_(q1,Heq)/2).

5) Update estimates for θ_(q1,Heq).

6) Go back to step 2 and repeat until derivative estimates are zero (orsatisfactorily near aero), or some other exit criteria is met.

Again, the nature of this gradient search technique is that the finalimbalance level will be limited to somewhat below the noise floor of thedesired signal, depending on convergence parameter μ. In general,smaller values for μ would allow convergence to a lower noise floor, butcause slower convergence. Schemes might be employed whereby μ adaptsfrom larger values to smaller values either with time, or as residualimbalance energy diminishes. As previously stated, this kind of adaptivealgorithm sometimes goes by the name “simulated annealing.”

Results of Simulation

As example of the techniques discussed, a simulation was run todemonstrate convergence to a solution where both LO feed-through and I/Qimbalance were mitigated. The results are illustrated in FIGS. 22A-D.

Parameters used in the simulation included:

M=4

RF center frequency=16.7 GHz

RF chirp bandwidth=1.0 GHz

pulse width=1 μs

video samples per pulse=2048

pulses integrated=64

video SNR=60 dB

Clearly, iterations can sink the problem energy into the noise floor,which can be lowered arbitrarily by integrating more pulses.Furthermore, integrating more pulses can also compensate a lesser videoSNR.

Hardware Hooks Required

If LO feed-through is adequately stable and constant with input signalfrequency from the QDWS, a single value for (t_(m)−t_(s)) can berequired to generate a video signal to facilitate estimation ofoffsetting DC inputs. This could be easily done with a short delay-line.If, however, the LO feed-through is QDWS frequency dependent, then somecontrol is required over the quantity γ_(T)(t_(m)−t_(s)). This is alsotrue for the generally frequency dependent I/Q imbalance.

The ideal situation would be the incorporation of a delay line togenerate the delay t_(s) such that t_(s)>T by enough time to allowswitching the radar hardware from transmit to receive. The delay t_(m)would be generated by initiating a second ‘receive’ pulse in the usualmanner. Consequently, adjusting the quantity (t_(m)−t_(s)) can beaccomplished by adjusting the ‘receive’ chirp delay t_(m). Both positiveand negative relative timings can be so generated. Note that theimportant quantity to adjust is γ_(T)(t_(m)−t_(s)), suggesting that fora fixed delay (t_(m)−t_(s)) chirp rate γ_(T) might be adjusted instead.

The embodiments and examples set forth herein are presented in order tobest explain the present invention and its practical application and tothereby enable those skilled in the art to make and utilize theinvention. However, those skilled in the art will recognize that theforegoing description and examples have been presented for the purposeof illustration and example only. The description as set forth is notintended to be exhaustive or to limit the invention to the precise formdisclosed. Many modifications and variations are possible in light ofthe above teaching without departing from the spirit and scope of thefollowing claims.

We claim:
 1. A method for adjusting waveform generation from a radarsystem employing QDWS and SSB mixer components, comprising the steps of:applying a rolling phase shift to the radar's QDWS signal, wherein saidrolling phase shift is demodulated in a receiver; and separatingimbalance energy from a desired signal in Doppler.
 2. The method ofclaim 1, further including the step of filtering the separated imbalanceenergy from the receiver leaving the desired signal.
 3. The method ofclaim 1, wherein said desired signal is provided as compensation forfrequency dependent errors transmitted through components affectingradar signal quality.
 4. The method of claim 1, wherein signalcorrection is provided as compensation for frequency dependent errorstransmitted through the QDWS.
 5. The method of claim 1, wherein signalcorrection is provided as compensation for frequency dependent errorstransmitted through the SSB mixer.
 6. The method of claim 3, furthercomprising the step of measuring the separated imbalance energy in thereceiver to thereby determine a degree of imbalance represented by saidimbalance energy.
 7. The method of claim 6, further comprising usingsaid degree of imbalance to determine calibration values that can to beprovided to the QDWS to cause signal correction.
 8. The method of claim7, wherein signal correction is provided as compensation for frequencydependent errors in the QDWS.
 9. The method of claim 7, wherein signalcorrection is provided as compensation for frequency dependent errors inthe SSB mixer.
 10. The method of claim 7, wherein signal correction isprovided as compensation for frequency dependent errors in componentsaffecting quadrature signal quality.
 11. A method for adjusting waveformgeneration in a radar, comprising the steps of: applying a rolling phaseshift to the radar's QDWS signal, wherein said rolling phase shift isdemodulated in a receiver; separating imbalance energy from a desiredsignal in Doppler; and filtering the separated imbalance energy from thereceiver leaving the desired signal.
 12. The method of claim 11, whereinsaid desired signal is provided as compensation for frequency dependenterrors transmitted through components affecting radar signal quality.13. The method of claim 11, wherein signal correction is provided ascompensation for frequency dependent errors transmitted through theQDWS.
 14. The method of claim 11, wherein signal correction is providedas compensation for frequency dependent errors transmitted through theSSB mixer.
 15. The method of claim 11, further comprising the step ofmeasuring the separated imbalance energy in the receiver to therebydetermine a degree of imbalance represented by said imbalance energy.16. The method of claim 15, further comprising using said degree ofimbalance to determine calibration values that can to be provided to theQDWS to cause signal correction.
 17. The method of claim 16, whereinsignal correction is provided as compensation for frequency dependenterrors in the QDWS.
 18. The method of claim 16, wherein signalcorrection is provided as compensation for frequency dependent errors inthe SSB mixer.
 19. The method of claim 16, wherein signal correction isprovided as compensation for frequency dependent errors in componentsaffecting quadrature signal quality.
 20. A method for adjusting waveformgeneration in a synthetic aperture radar system (SAR), comprising thesteps of: applying a rolling phase shift to the SAR's QDWS signal,wherein said rolling phase shift is demodulated in a receiver;separating imbalance energy from a desired signal in Doppler; filteringthe separated imbalance energy from the receiver leaving the desiredsignal; and measuring the separated imbalance energy in the receiver tothereby determine a degree of imbalance represented by said imbalanceenergy, wherein said degree of imbalance is further used to determinecalibration values that can to be provided to the QDWS to cause signalcorrection.
 21. The method of claim 20, wherein signal correction isprovided as compensation for frequency dependent errors in the QDWS. 22.The method of claim 20, wherein signal correction is provided ascompensation for frequency dependent errors in the SSB mixer.
 23. Themethod of claim 20, wherein signal correction is provided ascompensation for frequency dependent errors in components affectingquadrature signal quality.